Later Monday, at a meeting with pharmaceutical executives, Mr. Trump said that “we will continue to do exactly what we’re doing,” even as he hinted that he may announce new travel restrictions from a number of unnamed nations. Mr. Pence said that the Trump administration would screen all travelers on direct flights to the United States from Italy and South Korea, where thousands of coronavirus cases have appeared in recent weeks.

A senior administration official said that the White House had talked about a broader ban on travel, among other options, but that Mr. Trump had not made a decision.

Earlier in the day, Mr. Pence discussed the virus with the nation’s governors, and later joined the president in the meeting with pharmaceutical companies where Mr. Trump said he heard that a vaccine would be ready in three to four months. Dr. Fauci, who was in the room, clarified that deploying a vaccine was at least a year away.

The drug company executives told Mr. Trump that it would still take a year to 18 months to produce a viable vaccine in quantities for widespread use.

“That’s going to require testing periods,” Dr. J. Joseph Kim, the chief executive of Inovio Pharmaceuticals, said in an interview after the White House meeting. “Obviously we’re working at warp speed on this. Just think — we didn’t even have ‘coronavirus’ in our vocabulary until early January.”

Dr. Kim said his firm would begin human trials of a vaccine next month, but it would take until the end of the year or early next year to be ready for the broader public.

“We can produce as much as one million doses by the end of this year using our existing capacity and resources,” he said. “But we need help from the U.S. government and resources it could bring to scale. If we have a successful vaccine, we need to make hundreds of millions of doses.”