In media appearances and talks with investors, Pfizer’s chief executive nearly always mentions a word that is so politically perilous, most of his competitors shy away from it: October.

“Right now, our model — our best case — predicts that we will have an answer by the end of October,” the chief executive, Dr. Albert Bourla, told the “Today” show earlier this month. In other interviews, he has said he expected a “conclusive readout” by then, with an application for emergency authorization that could be filed “immediately.”

Dr. Bourla’s statements have put his company squarely in the sights of President Trump, who has made no secret of his desire for positive vaccine news to boost his chances on Election Day, Nov. 3. “We’re going to have a vaccine very soon. Maybe even before a very special date,” Mr. Trump said recently.

And yet by all other accounts, the idea that it will be ready in October is far-fetched. Even if the vaccine shows promising signs in clinical trials — still a big if — the company will not have collected enough data by then to say with any statistical confidence that it is safe and effective.

By repeating a date that flies in the face of most scientific predictions, Dr. Bourla is making a high-stakes gamble. If Pfizer puts out a vaccine before it has been thoroughly tested — something the company has pledged it will not do — it could pose a major threat to public safety. The perception matters, too: If Americans see the vaccine as having been rushed in order to placate Mr. Trump, many may refuse to get the shot.

But there is a significant upside, to the tune of billions of dollars, in being first to the U.S. market with a vaccine. And staying in the president’s good graces — particularly when he keeps talking about ways to lower drug prices — might not be a bad thing for a company that brought in nearly $40 billion in 2019 from sales of high-priced, brand-name drugs.

“There’s a huge financial advantage to being first out of the gate,” said Dr. Megan Ranney, an associate professor of emergency medicine and public health at Brown University. She was one of 60 public health officials and others in the medical community who signed a letter to Pfizer urging it not to rush its vaccine.

And given the White House’s persistent efforts to interfere in the decisions of federal health agencies, some scientists fear a vaccine approval could come under similar pressure.

“What I worry about is that the politics or the financial gain may drive earlier release than is scientifically appropriate,” Dr. Ranney said.

Pfizer will not be anywhere near completion of its clinical trial by the end of October, according to a company spokeswoman. When Dr. Bourla referred to a “conclusive readout” next month, she said, he meant that it’s possible the outside board of experts monitoring the trial would have by that date found promising signs that the vaccine works.

Pfizer’s trial blueprints allow the panel to look at the data after just 32 volunteers have become ill with Covid-19. Even if most of those cases were in the placebo group, regulators at the Food and Drug Administration would likely need data from more volunteers before making a decision on authorization, according to people familiar with the government’s vaccine approval process.

The F.D.A. has also told vaccine makers that they will need to track at least half of the patients’ safety data for two months before the agency will grant emergency access. That would push the earliest possible date into at least November.

In a statement, Pfizer said that it would continue to collect data on trial volunteers’ health for two years, and that it planned to submit two months of safety data to the agency “to help inform F.D.A.’s ultimate determination of authorization or approval.”

In public interviews, government health officials have refuted the October date. Both Dr. Moncef Slaoui, the chief scientific adviser to Operation Warp Speed, the federal effort that has awarded billions of dollars to vaccine makers, and Dr. Anthony S. Fauci, the nation’s top infectious disease doctor, have said October was unlikely.

Pfizer’s leading competitors in the vaccine race, Moderna and AstraZeneca, have been more vague about timing, saying they expect something before the end of the year. In a recent interview, Moderna’s chief executive, Stéphane Bancel, said: “October is possible, because very few things in life are impossible.” The better word, he said, is “unlikely.”

“They will not get approved before the election,” said Ronny Gal, an analyst with the Wall Street firm Bernstein. “The timing just doesn’t work.”

One key to understanding how Pfizer vaulted to first place in the vaccine race is found in its trial blueprints, known as a protocol.

Even though Moderna and Pfizer began their trials on the same day, Pfizer’s is “built for speed,” as one Wall Street firm, SVB Leerink, described it.

Participants in Pfizer’s trial are given two doses of a vaccine 21 days apart, whereas those in Moderna’s wait 28 days in between. Pfizer begins looking for sick volunteers seven days after the second dose, whereas Moderna does so at 14 days. And Pfizer’s plan allows an outside review panel to look at early data after just 32 volunteers have become ill with Covid-19. Moderna’s plan doesn’t allow for a first peek until 53 cases.

Some experts have speculated that Pfizer’s volunteers happen to be in places where the virus is spreading quickly, making it more likely that they would have been exposed to the virus and put the vaccine to the test. While Moderna’s trial is being conducted only in the United States, Pfizer’s is international, with locations set up or planned in the United States, Brazil, Turkey and Argentina.

In Argentina, which has seen a surge of coronavirus infections since August, 33,000 volunteers quickly signed up for about 4,500 slots, said Dr. Fernando Polack, the head researcher of the Pfizer study that is being carried out at the Hospital Military Central in Buenos Aires.

“We have a lot of years of experience in these kinds of trials, and we’ve never seen these numbers before,” Dr. Polack said in an interview. He declined to comment on whether he was aware of any Covid-19 cases among volunteers.

Across all of its trial sites, Pfizer said, 24,000 volunteers have received the second dose of the vaccine, more than half of the 44,000 that are expected to enroll.

Despite Pfizer’s confident pronouncements, a win is not guaranteed. Around 15 percent of vaccines in late-stage trials do not make it to approval.

Dr. Mark Goldberger, an infectious disease expert at the Global Antibiotic Research and Development Partnership and a former F.D.A. official, noted that the agency has scheduled a meeting of outside experts to discuss experimental coronavirus vaccines on Oct. 22.

“It would take an enormous effort to get the data by then,” he said. “It’s within the realm of possibility, but there are things that could trip them up — a manufacturing issue, a safety issue, the efficacy is not as good as they expect.”

Taking first place in the vaccine race isn’t just about bragging rights, although the public-relations boost from developing a vaccine that could liberate the world from a deadly pandemic can’t be understated.

Mr. Gal, the Bernstein analyst, said that Pfizer could win a significant commercial edge by taking control of the early market for a coronavirus vaccine. Unlike Johnson & Johnson and AstraZeneca, which have said they will not profit from their products during the pandemic, Pfizer has made no such promises.

It also has not taken federal money to develop its vaccine, instead signing a $1.95 billion deal to sell the first 100 million doses of its vaccine to the U.S. government. Dr. Bourla has said that the company didn’t accept federal investment in its research and development so that the government wouldn’t be able to control the price of the vaccine later on.

“We didn’t take money so that we don’t have any restrictions,” Dr. Bourla said on Sept. 16 at an event organized by the investment bank J.P. Morgan. “Because I can imagine, if you take money, then you have to — a little bit to explain about your pricing.”

While smaller companies like Moderna have already benefited financially by rising share prices, Pfizer has not seen a similar stock bump. Mr. Gal has estimated the initial wave of vaccination as an $18 billion market. “So to the extent that they are able to capture any material amount of it, even for a company like Pfizer, it’s not a small thing,” he said.

Pfizer also has more at stake when it comes to doing business with the federal government. Moderna does not sell any approved products, but Pfizer has a broad portfolio of vaccines and drugs. The federal government is both a major customer — paying for treatments and vaccines through Medicare — and makes weighty decisions about everything from drug approvals to drug-pricing policy.

Even as Mr. Trump has urged the speedy approval of a vaccine, he has employed harsh rhetoric against the industry. Earlier this month, he issued an executive order aimed at lowering the cost of some drugs by tying them to prices in other countries. The measure was strongly opposed — and is likely to be challenged in court — by the drug industry.

Earlier this month, the president singled out Pfizer for special praise, describing it as a “great company,” and saying he had spoken to Dr. Bourla, who was a “great guy.”

Dr. Bourla, who has spent much of the past month giving interviews to media outlets and appearing on industry panels, has insisted the company’s accelerated timeline had nothing to do with politics. “We have politicians or journalists speaking about efficacy or safety about medicines which, of course, is not appropriate,” Dr. Bourla said at the company’s investor day on Sept. 15. “The scientists should be having these discussions.”

Ultimately, Pfizer’s strategy may be about managing the public’s expectations, said Brandon Barford, a partner at Beacon Policy Advisors, a research firm. Pfizer could now explain any delay past October by “saying, ‘We’re being extra cautious.’ And you get kudos for it.”

But if the opposite occurs, and Pfizer is seen to be pushing a vaccine before it is ready, the “potential fallout is enormous,” said Dr. Ranney, of Brown University. “We cannot afford to have a vaccine released for Covid-19 that is either unsafe or ineffective.”

Sharon LaFraniere, Daniel Politi, Noah Weiland and Katherine J. Wu contributed reporting.