No, the executives told him, probably not.

Apparently trying to manage Mr. Trump’s expectations, Dr. Leonard Schleifer, the chief executive of Regeneron, a biotechnology company, delicately suggested that an essential truth about scientific research is that success is cultivated from a large mound of failed experiments.

“We have a group of people around this table, myself included, who are in an industry where optimism is an essential part of the tool kit,” Dr. Schleifer said. “But realism is that, you know, 95 percent of what we all work on doesn’t go too far. So we — that’s why it’s so important to have so many different approaches.”

Like other drugs, vaccines go through phases of rigorous trials, first on animals, then on humans, to determine safety and efficacy before they can be approved and disseminated widely. As the executives pitched the efficiency of their companies’ vaccine development processes, they discussed how quickly a test vaccine could be ready for initial trials. Some suggested three months, others two.

Apparently Mr. Trump thought they were referring to the time frames for producing a completed vaccine. He turned to Dr. Anthony S. Fauci, director of the National Institute of Allergy and Infectious Diseases, and said:

“I’ve heard very quick numbers — a matter of months — and I’ve heard pretty much a year would be an outside number. So I think that’s not a bad — that’s not a bad range. But if you’re talking about three to four months, in a couple of cases, and a year in other cases — wouldn’t you say, Doctor, would that be about right?”

Dr. Fauci turned to the executives and officials around the table, and repeatedly tapped the polished wood surface for emphasis. “Would you — would you make sure you get the President the information that a vaccine that you make and start testing in a year is not a vaccine that’s deployable. So he’s asking the question, ‘When is it going to be deployable?’ And that is going to be, at the earliest, a year to a year and a half, no matter how fast you go.”

The executives said they could have therapeutic medications to ease symptoms ready more quickly, which seemed to assuage the president.

Later that night, at a rally in Charlotte, N.C., Mr. Trump was still heralding the quick arrival of a vaccine to solve the coronavirus problem.

“Today we met with the big great pharmaceutical companies and they’re really working hard and they’re working smart,” he said. “And we had some — we had a great meeting today with a lot of the great companies and they’re going to have vaccines, I think relatively soon.”

Jack Begg and Dahlia Kozlowsky contributed research.