President Trump said amid the coronavirus outbreak that there may be more travel restrictions coming, "from certain countries where they're having more of a breakout." He did not specify which countries, however. On Saturday, Mr. Trump announced restrictions on visitors from Iran and further screening of people traveling from South Korea and Italy. Monday's disclosure came during a meeting he and Vice President Mike Pence were having with top pharmaceutical company representatives at the White House, as the federal government and private industry scramble to come up with a vaccine and treatment for coronavirus as quickly as possible.

The president went around the table with the executives and listened to them make pitches on formulating coronavirus vaccine. He listened to pitches and asked about the timing for manufacturing an effective vaccine.

The CEO of Moderna, Stéphane Bancel, told Mr. Trump that he expected to begin phase-one testing of a vacccine and was just waiting for a green light from the National Institutes of Health. After that, Bancel said it would take a few months to get human data, before phase-two testing could begin.

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Mr. Trump asked him, "So, you're talking over the next few months you think you could have a vaccine?"

Bancel replied, "Correct," then he added, "Phase two."

Flanked by Vice President Mike Pence (L) and Secretary of Health and Human Services Alex Azar, President Trump leads a meeting with the White House Coronavirus Task Force and pharmaceutical executives in Cabinet Room of the White House on Monday, March 2, 2020, in Washington, D.C. Getty

At this point, Anthony Fauci, director of the National Institute of Allergy and Infectious Diseases, jumped in to say, "You won't have a vaccine — you'll have a vaccine to go into testing."

Mr. Trump said, "And how long will that take?"

Phase two "will take a few months before we can go into to phase three," Bancel said.

"Like I've been telling you, Mr. President, a year to a year and a half," Fauci said.

Another participant, Dr. Leonard Schleifer, CEO of Regeneron Pharmaceuticals pointed out that Fauci's caution was well-founded. "There's precedent for vaccines to actually make diseases worse. You don't want to rush and treat a million people and find you've made 900,000 people worse."

On Sunday night, the second confirmed death from coronavirus in the U.S. was reported, adding to an earlier death also in Washington state. Experts fear the number of infected people in the U.S. far exceeds the number health officials have officially confirmed.

The president told those present he's considering more travel restrictions from countries where the virus is rampant, but declined to say which countries. The administration announced further restrictions on travel from Iran on Saturday.

These are the executives the president met Monday:

Emma Walmsley, CEO, GlaxoSmithKline

Dr. J. Joseph Kim, CEO, Inovio Pharmaceuticals

Stéphane Bancel, CEO, Moderna

Dr. Leonard Schleifer, CEO, Regeneron Pharmaceuticals

Stan C. Erck, CEO, President, and Director, Novorax

Daniel Menichella, CEO, CureVac

David O'Day, Chairman of the Board of Directors and CEO, Gilead

Dr. Paul Stoffels, Vice Chairman of the Executive Committee and Chief Scientific Officer, Johnson and Johnson

Dr. Mikael Dolsten, Chief Scientific Officer and President Worldwide Research, Development and Medical, Pfizer Inc.

John Shiver, Senior Vice President, Global Vaccine Research and Development, Sanofi

The administration is trying to reassure the public that the risk to most Americans remains "low," even as more cases continue to be reported. Pence, who has been tasked by the president to lead the efforts against the virus, and Ambassador Deborah Birx, a global health official and physician who will be the coronavirus response coordinator in Pence's office, briefed reporters on their efforts while the president traveled to a campaign rally.

The Department of Health and Human Services has launched an investigation into manufacturing problems with of the initial batch of coronavirus testing kits. The Food and Drug Administration says the issue has since been resolved and that the test kits currently being distributed are safe and effective.