Joining Mr. Trump at the briefing, Vice President Mike Pence announced that the State Department would issue its highest-level warning, known as a “do not travel” alert, for areas of Italy and South Korea most affected by the virus. Mr. Pence said the United States was also banning all travel from Iran, and barring entry to any foreign citizen who has visited that country in the last 14 days.

In response to a question, Mr. Trump also said he was “very strongly” considering new travel restrictions along America’s southern border to contain the coronavirus. But he later seemed to play down the possibility, saying “this is not a border that seems to be much of a problem right now.”

He also disputed criticism that he had trivialized the coronavirus during a campaign rally Friday night in South Carolina, in which he said of Democrats: “This is their new hoax.” The comment drew swift condemnation from his political rivals.

“The ‘hoax’ was used with respect to Democrats and what they were saying,” Mr. Trump said on Saturday.

Mr. Trump said that 22 people were currently ill with the virus, including four he said are “very ill.” In fact, 70 cases have been reported in the country, though just one of them fatal. Most could be explained by overseas travel or contact with someone who had been ill. This week, though, four new cases, in California, Oregon and Washington, were the first in the United States where the cause was unknown — a sign, experts warned, that the virus, which has killed more than 2,800 people worldwide, now threatens everyone.

Several officials at the briefing, including Dr. Redfield and Dr. Anthony S. Fauci, director of the National Institute of Allergy and Infectious Diseases, warned Americans to gird themselves for more bad news.

“We need to prepare for further challenges, and we will have them,” Dr. Fauci said. “You will hear about additional cases that will be coming on. You should not be surprised by that.”