U.S. lawmakers struggle with how to get a Covid aid package passed over an immigration fight.

Lawmakers on Capitol Hill continue to struggle with how to advance a pandemic aid package amid an election-year dispute over immigration, even as Congress barrels toward swift approval of $40 billion in emergency aid for Ukraine.

The White House has repeatedly pushed for a $22.5 billion package for vaccines, therapeutics and other treatments at home and abroad, warning about the dire consequences of inaction. But a smaller package — whittled down to $10 billion as Republicans demanded that any new coronavirus spending should be paid for — has been stalled in the Senate as conservatives push to include language that maintains immigration restrictions at the country’s land borders.

Some Senate Democrats are now signaling a willingness to hold a vote on an amendment that would require the administration to reinstate an emergency public health order that has restricted immigration since the beginning of the pandemic, if it allows for passage of the coronavirus aid package. The Biden administration has said it will lift the order, known as Title 42, on May 23.

The shift comes after President Biden told Democratic leaders this week to drop efforts to combine the Ukraine aid and the pandemic funds in a single measure, reflecting concerns from both parties that the snarled politics of the coronavirus package could delay more assistance for Ukraine in its fight against Russia.

“I think there’s a growing willingness to bring that up and have the amendment votes necessary to get it to a final vote,” Senator Chris Murphy, Democrat of Connecticut, said Tuesday, referring to the legislation. It is unclear, however, whether that would derail the package given that liberals remain adamant about ending the Title 42 policy, which gives border officials the authority to quickly expel undocumented migrants, even if they are seeking asylum.

Liberals in both chambers have expressed some quiet frustration that the pandemic aid has become hamstrung by demands that it be paid for, as well as by the immigration fight, while the emergency Ukrainian aid cleared the House on Tuesday less than 12 hours after the legislation became public.

The White House is frustrated as well. Mr. Biden is convening a global Covid-19 summit on Thursday that is aimed at revitalizing the lagging global response. The $10 billion compromise package excludes any funding for helping other countries fight the pandemic.

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, in order to secure the coronavirus aid. “Schumer has tried to get us into a circumstance where that’s not called,” said Senator Richard J. Durbin of Illinois, the No. 2 Democrat, referring to a vote on Title 42, but “there’s some things he just can’t achieve.”

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

It remains unclear whether the package will grow from a $10 billion deal for domestic pandemic spending struck between Mr. Schumer and other 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.

Senator Mitt Romney, Republican of Utah and a key negotiator behind the $10 billion measure, rejected the possibility of supporting a larger package, saying on Tuesday, “Nope, we’ve got a deal at 10. Let’s get it done.”

“I’m just anxious to get the vote on the Covid package as soon as we can,” he added.

Sheryl Gay Stolberg and Eileen Sullivan contributed reporting.