In April, with hospitals overwhelmed and much of the United States in lockdown, the Department of Health and Human Services produced a presentation for the White House arguing that rapid development of a coronavirus vaccine was the best hope to control the pandemic.

“DEADLINE: Enable broad access to the public by October 2020,” the first slide read, with the date in bold.

Given that it typically takes years to develop a vaccine, the timetable for the initiative, called Operation Warp Speed, was incredibly ambitious. With tens of thousands dying and tens of millions out of work, the crisis demanded an all-out public-private response, with the government supplying billions of dollars to pharmaceutical and biotechnology companies, providing logistical support and cutting through red tape.

It escaped no one that the proposed deadline also intersected nicely with President Trump’s need to curb the virus before the election in November.

The ensuing race for a vaccine — in the middle of a campaign in which the president’s handling of the pandemic is the key issue after he has spent his time in office undermining science and the expertise of the federal bureaucracy — is now testing the system set up to ensure safe and effective drugs to a degree never before seen.

Under constant pressure from a White House anxious for good news and a public desperate for a silver bullet to end the crisis, the government’s researchers are fearful of political intervention in the coming months and are struggling to ensure that the government maintains the right balance between speed and rigorous regulation, according to interviews with administration officials, federal scientists and outside experts.

Even in a less politically charged environment, there would be a fraught debate about how much to accelerate the process of trials and approval. The longer that vaccines are tested before being released, the likelier they are to be safe and effective.

But with 1,000 people dying each day in the United States, schools finding it difficult to reopen and the deep recession inflicting economic pain across the country, the desire to find a way to return to normal life is powerful and transcends partisan politics and borders. On Sunday, Russia announced that it planned to start a nationwide inoculation campaign in October with a vaccine that had yet to complete clinical trials, the latest evidence of the global potential for cutting corners.

Despite concerted efforts by the Trump administration and a bevy of pharmaceutical companies it is working with, the original October target has slipped, with the administration now pushing to have hundreds of millions of doses available by the end of the year or early 2021.

But experts inside and outside the government still say they fear the White House will push the Food and Drug Administration to overlook insufficient data and give at least limited emergency approval to a vaccine, perhaps for use by specific groups like front-line health care workers, before the vote on Nov. 3.

“There are a lot of people on the inside of this process who are very nervous about whether the administration is going to reach their hand into the Warp Speed bucket, pull out one or two or three vaccines, and say, ‘We’ve tested it on a few thousand people, it looks safe, and now we are going to roll it out,’” said Dr. Paul A. Offit of the University of Pennsylvania, who is a member of the Food and Drug Administration’s vaccine advisory committee.

“They are really worried about that,” he added. “And they should be.”

Mr. Trump relentlessly touts progress toward a vaccine, raising hopes of quick approval. Touring a North Carolina biotechnology lab last week, he vowed to “deliver a vaccine in record time.” In a tweet last month, he explicitly tied vaccines to his re-election hopes.

On a campaign call with supporters in Pennsylvania on Sunday evening, Mr. Trump said the “F.D.A. has been great, at my instruction,” and he again raised hopes of rapid progress.

“We expect to have a vaccine available very, very early before the end of the year, far ahead of schedule,” he said. “We’re very close to having that finalized.”

The president’s son-in-law and senior adviser, Jared Kushner, who is helping to steer the re-election campaign from the White House, is a regular participant in meetings of a board formed to oversee the vaccine effort.

While White House officials do not specifically mention the election during the board’s discussions, people familiar with the conversations say they ask regularly about October, a date that hangs over the effort. Trump campaign advisers privately call a pre-election vaccine “the holy grail.”

The Food and Drug Administration’s approval of a new vaccine is typically an exhaustive process, where agency employees meticulously go through data from clinical trials to review whether the vaccine is both safe and effective. The threshold for approving vaccines is typically higher than it is for therapeutic drugs because they will be used in millions of otherwise healthy people, meaning that even rare side effects could affect many more people than a drug that treats a specific illness.

An independent advisory panel of outside experts also weighs in, and while the agency has the power to make its own decision, it typically follows the advice of its outside panels. The Food and Drug Administration’s senior regulator has the power to approve or deny vaccines for emergency use, but that decision could be overridden by the agency’s top leaders, or by the secretary of health and human services.

White House officials said that Mr. Trump would not distort the vaccine review process to help his campaign. “The rapid research, development, trials and eventual distribution of a Covid-19 vaccine is emblematic of President Trump’s highest priority: the health and safety of the American people,” said Judd Deere, a White House spokesman. “It has nothing to do with politics.”

Dr. Anthony S. Fauci, the director of the National Institute of Allergy and Infectious Diseases, told lawmakers on Friday that he remained “cautiously optimistic that we will have a vaccine by the end of this year and as we go into 2021.”

Dr. Stephen Hahn, the commissioner of the Food and Drug Administration, has not ruled out emergency approval of a vaccine.

“We would consider using an emergency use authorization if we felt that the risks associated with the vaccine were much lower than the risks of not having a vaccine,” he told The Journal of the American Medical Association in an online interview.

He also said regulators would certify that any vaccine would meet the agency’s rigorous standards, adding, “My job as commissioner is to make sure to the fullest extent possible that any pressure that comes to the agency is not reflected downward” onto regulators and scientists studying the vaccines.

At the same time, a senior administration official refused to promise that any emergency approval of a vaccine would be vetted through the Food and Drug Administration’s outside advisory panel of experts, scheduled to meet on Oct. 22.

Operation Warp Speed got its start in April, the brainchild of Dr. Peter Marks, a pencil-thin, bespectacled physician who leads the regulatory unit at the Food and Drug Administration that approves vaccines and therapies.

A “Star Trek” fan, Dr. Marks named the initiative Warp Speed and pitched it in an April 10 phone call to Alex M. Azar II, the secretary of health and human services, who quickly embraced it. In a follow-up phone call a few days later, according to a person familiar with the discussions, several health officials said the October deadline was unrealistic; over the next few months, officials began publicly citing the end of the year or early 2021 as a target.

With his job on the line, Mr. Azar, the target of Mr. Trump’s wrath over the virus, was especially eager to prove his worth to the White House. He teamed up with Defense Secretary Mark T. Esper, whose department has long experience with vaccine development and distribution to protect troops. An expert in complex logistics, Gen. Gustave F. Perna, became the operation’s chief operating officer.

Mr. Kushner, Dr. Deborah L. Birx, the White House coronavirus coordinator, and others interviewed Dr. Moncef Slaoui, a pharmaceutical industry veteran, and orchestrated his appointment as chief scientific adviser despite concerns within the Food and Drug Administration about conflicts of interest because of his financial ties to two companies that are developing a vaccine. Rather than being bothered by the conflict, Mr. Kushner and others reasoned that it took someone with such industry experience to oversee the effort.

Dr. Slaoui resigned from the board of Moderna, which has received nearly $1 billion in federal support to develop a vaccine. But as of May he still had nearly $10 million of stock in GlaxoSmithKline, a partner with the French drugmaker Sanofi, which last week signed a $2.1 billion agreement to produce 100 million doses. Dr. Slaoui, who is working on a $1 contract, cleared an ethics review by the Department of Health and Human Services and has said he is determined to avoid any conflict.

Shortly after Dr. Slaoui’s appointment, Dr. Marks resigned from the project he conceived and returned full-time to his post as a senior regulator at the Food and Drug Administration, where he will be the key decision maker on whether a vaccine merits approval.

The administration has conducted the vaccine hunt with a focus lacking in much of the rest of its pandemic response. Contracts have been executed at a brisk pace. Mobile trailers have been speedily delivered for experimental doses to be administered. When a company was short on needles, the Pentagon dispatched planes to deliver supplies within 48 hours.

The pharmaceutical companies are reporting the results of their trials at regular intervals, accelerating the review process. With the government paying much of the cost, the companies are beginning the process of manufacturing millions of doses of vaccine essentially on spec so that they can be distributed quickly if they secure approval.

The process has moved at a remarkable clip. Two vaccine candidates, one developed by Moderna in conjunction with Dr. Fauci’s institute and another by Pfizer, last week began Phase 3 trials, the final stage of clinical experimentation. Others are expected soon.

In Mr. Azar’s conference room at the Department of Health and Human Services headquarters, Mr. Kushner and Dr. Birx join meetings with Mr. Azar, Mr. Esper and others. Mr. Kushner repeatedly pushes the group to move faster and has deputized two close associates, Brad Smith and Adam Boehler, to press the case.

The team has sought to ensure that a variety of different types of potential vaccines are being pursued to increase the chances that at least one will work. Dr. Birx has been interested in what is known as a subunit protein vaccine, and at one point called executives at the biotechnology company Genentech and asked what they could do. (Warp Speed is now working with two companies pursuing that type of vaccine.)

Mark Meadows, the White House chief of staff, also talks with pharmaceutical executives. People briefed on the discussions say the White House has also pushed for progress by the fall on therapeutics — drugs to treat people who fall ill to the disease — including the possibility of an emergency use authorization for one or more of those drugs. Late last month, Mr. Trump called the chief executive of Regeneron Pharmaceuticals to check on the progress of a potential antibody treatment.

Career officials have assured Dr. Hahn that they would stand behind him to head off any vaccine decision not based on science. But Dr. Hahn already lost a measure of credibility with the scientific community for approving the emergency use of hydroxychloroquine and chloroquine, two anti-malaria drugs promoted by the president as treatments for the coronavirus over the objections of his public health advisers. The Food and Drug Administration later revoked the authorization, concluding the risks outweighed the benefits.

Scientists have argued that it would be unwise to cut corners on a vaccine that is to be injected into some 300 million Americans, adding that a failed effort would fuel public distrust of vaccines generally.

But a senior White House official, who discussed the matter on the condition of anonymity, said that it would also be unethical to withhold an effective vaccine for an extra three or four months while more people died just to check the boxes of a more routine trial process.

Michael R. Caputo, a spokesman for Mr. Azar, said October was not the goal.

“Everybody at H.H.S. hopes Operation Warp Speed will achieve 300 million doses of a safe and effective Covid vaccine for Americans by January 2021,” he said. “We know that’s optimistic. I have never heard mention of any other timeline, and certainly not from the secretary.”

“Careless talk about career F.D.A. regulators somehow approving an unsafe and ineffective vaccine just for politics only undermines confidence in the public health system,” he added.

It is not clear that a vaccine approval shortly before the election would be an “October surprise” sufficient to alter the outcome of the vote. An announcement could give Americans hope that the end is in sight. But some Republican strategists said that it might not help Mr. Trump because his opponent, former Vice President Joseph R. Biden Jr., the presumptive Democratic nominee, would surely continue the vaccine process if elected.

“Does it turn everything around for him politically? I don’t know,” said Sarah Longwell, a conservative strategist and prominent Republican opponent of Mr. Trump who regularly conducts focus groups and has found that public attention is more focused on government relief checks and school reopenings.

“If the vaccine is an October surprise, there’s a lot of other things that are cutting against” it as a game-changer, she said.

The drug companies find themselves caught in the middle. While eager to bring products to market as quickly as possible, they face risks in moving too quickly in order to fit an election calendar, analysts said.

“They are acutely aware of the political dynamic here,” said Rob Smith, the director of Capital Alpha Partners, a research firm. A vaccine that flopped would jeopardize their broader business, he said, and it would not make sense “to take a huge reputational risk not just for your vaccine but for all the products across your portfolio to benefit the president politically.”

Dr. Fauci has expressed confidence that the system will hold.

“Historically, the F.D.A. has based their decisions on science,” he told a House committee last week. “They will do so this time also, I am certain.”

Maggie Haberman contributed reporting and Kitty Bennett contributed research.