“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 that his company would begin human trials of a vaccine next month, but that 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.”

But Dr. Kim said he was upbeat after the meeting about the prospect of help from the government and collaboration with other drug companies. “I walked away feeling very optimistic about our own work but also our comrades in arms,” he said.

Democrats remained unimpressed by the Trump administration’s efforts. Senator Chuck Schumer of New York, the Democratic leader, took to the Senate floor to assail Mr. Trump for his handling of the crisis.

“Even now, President Trump seems to be spending more of his time blaming the media, and blaming the Democrats, than being constructive,” Mr. Schumer said. “In fact, he blames everyone not named Donald Trump. The president is downplaying — he is downplaying — the threat of coronavirus to a dangerous degree, and his chief of staff, amazingly, said to Americans, ‘Turn off your televisions.’”

Mr. Trump’s meetings on the issue came before he flew to Charlotte, N.C., for a campaign rally on Monday night. He told reporters before leaving that it remained “very safe” for him and his Democratic opponents to hold such large campaign events even as businesses and nonprofit groups were canceling large conventions.