And $10 billion, administration officials say, will not be enough to address the domestic need for vaccines, drugs and therapeutics if, as many experts expect, infections surge in the fall and winter months. The White House has said it is preparing for 100 million Americans to be infected during those months; the figure is based on outside models. Still, the White House has been preparing for a scaled-down vaccination campaign should Congress not approve any money.

Senator Chuck Schumer of New York, the majority leader, declined on Tuesday to say whether he would agree to a Title 42 amendment vote, telling reporters that he would wait to see what legislation the House approves in the coming days and slamming Republicans for their demands.

“Our Republican friends should not be blocking Covid legislation,” he said. “We don’t know what they might throw in the way. We don’t even know if they want to pass it.”

Other Democrats acknowledged that they may need to vote on the immigration amendment, which has support from both Republicans and centrist Democrats, to secure the coronavirus aid. “Schumer has tried to get us into a circumstance where that’s not called,” Senator Richard J. Durbin of Illinois, the No. 2 Democrat, said Tuesday, referring to a vote on Title 42, but “there’s some things he just can’t achieve.”

Multiple lawsuits have been filed about various aspects of the public health rule, so legislation may not necessarily be the only way it would stay in place beyond May 23.

“I don’t think Title 42 has anything to do with Covid,” Senator Bob Menendez, Democrat of New Jersey, said Wednesday, criticizing Republicans for potentially putting “the health security of the nation at risk.”

It remains unclear whether the package will grow from a $10 billion deal for domestic pandemic spending struck between Mr. Schumer and Republicans, as some Democrats have pushed to revive aid for the global vaccination effort and other money that was left out. Republicans have objected to increasing the amount.