Congress adjourned for a two-week recess last week without voting on the package, which would provide more funding for treatments, testing and vaccines.

“We need this Covid funding,” Jen Psaki, the White House press secretary, said on “Fox News Sunday.” “And it’s really a huge disappointment to us that the Congress has left for two weeks without passing this into law.”

A small surge of coronavirus infections has swept through the halls of Washington over the last week. Dozens tested positive, including several administration officials and members of Congress, after attending the Gridiron Club and Foundation’s annual dinner on April 2.

President Biden did not attend the dinner, but his sister, who did, tested positive; Mr. Biden appeared, maskless, at several public events last week, raising concerns about whether he might have contracted the virus.

On Sunday, Dr. Fauci defended the procedures that are in place to protect the president.

“He’s fully vaccinated,” Dr. Fauci said. “He’s doubly boosted, and most of the time, people who get anywhere near him need to be tested. So we feel the protocol is a reasonable protocol.”

Dr. Fauci stressed that the seriousness of coronavirus infections should not be discounted and could still result in significant illness and long Covid, even if they don’t lead to hospitalization. But, he said, people need to make their own decisions about the risks they’re willing to take.

“This is not going to be eradicated, and it’s not going to be eliminated,” he said. “And what’s going to happen is that we’re going to see that each individual is going to have to make their calculation of the amount of risk that they want to take in going to indoor dinners and in going to functions.”