WASHINGTON — An election-year dispute over immigration policy emerged on Tuesday as the latest obstacle to quick approval of a $10 billion coronavirus response bill, as Senate Republicans refused to advance the measure without a vote to keep in place pandemic-era border restrictions that President Biden has moved to lift.

While lawmakers in both parties have said they support the money for vaccines, testing and therapeutics, Republicans blocked action on it on Tuesday, insisting that the chamber first vote to maintain the immigration policy, known as Title 42, which has restricted immigration at U.S. land borders since the beginning of the pandemic.

Democratic leaders declined to hold such a vote, which would expose deep divisions in their ranks over the Biden administration’s decision to wind down the order, and jeopardize passage of the broader package. It could also put some of their members at risk of a backlash from voters at a time when Republicans have tried to portray them as lax on immigration.

Senator Chuck Schumer of New York, the majority leader, said the legislation “should not be held hostage to extraneous unrelated issues,” and instead tried to push ahead with the funding package. The move failed on a vote of 52 to 47, with one Democrat, Senator Ron Wyden of Oregon, joining all Republicans in opposition.

“There’s still some time — I hope my Republican colleagues change their tune quickly,” Mr. Schumer said after changing his vote to “no,” a maneuver that allows him to move to reconsider the bill. The package would allocate at least $5 billion for therapeutics, as well as money for testing and vaccines.

While the Senate can try to take up the measure again, the unsuccessful vote on Tuesday signaled that final passage of aid that the Biden administration has said is urgently needed may slip until later this month, when Congress returns from a two-week break. Both chambers are scheduled to leave Washington at the end of the week.

“I’d like to move as quickly as possible,” said Senator Mitt Romney, Republican of Utah, who helped lead negotiations over the plan but opposed moving forward without a commitment to vote on the immigration issue and others. “Usually things have kind of a normal life cycle, and you’d hate to have them delayed by two weeks.”

Senate Democrats are also pushing to confirm Judge Ketanji Brown Jackson to the Supreme Court as early as Thursday, further complicating the prospects of salvaging a quick vote on the pandemic money. Any agreement to speed it through the Senate would require the support of all 100 senators, and Republicans said they were unwilling to do so without guaranteed votes on proposed additions.

“It will require, I think, several amendments in order to get across the floor,” said Senator Mitch McConnell of Kentucky, the minority leader. “So if the majority leader tries to move forward, we’ll have to have a discussion about a reasonable amendment process.”

While Republicans have pushed for other amendments, the most politically fraught provision they hoped to put to a vote would bar the Biden administration from ending the Trump-era pandemic restrictions at the border, which have granted officials the ability to turn away migrants, including those seeking asylum.

In announcing plans on Friday to end the order in late May, the Centers for Disease Control and Prevention cited “current public health conditions and an increased availability of tools to fight Covid-19.” But Republicans argued that it was contradictory for the administration to implement that policy change while asking Congress to approve more pandemic aid.

“Getting that money replenished is significantly complicated by announcing in the same week that you’re going to determine that Section 42 of this effort to fight this disease is no longer necessary because the disease is no longer the same kind of problem it was,” said Senator Roy Blunt, Republican of Missouri. “It’s clearly a big issue with our members.”

Multiple Democrats, including senators facing tough re-election contests in November, have said they have concerns about the decision to roll back the policy, given that it is expected to draw thousands more migrants to the southwestern border, where stations are already backed up and overcrowded. On Tuesday, Senator Jon Tester of Montana became the latest Democrat to ask the administration to provide “a comprehensive plan on how it will address the increased strain on our immigration system and increased security needs.”

Some of those Democrats, including Senators Mark Kelly of Arizona and Raphael Warnock of Georgia, declined to say whether they would support a Republican amendment, given that text has not been released. But if even one Democrat were to back it, the amendment could pass the evenly divided Senate. That would most likely kill the funding package in the House, where most Democrats have cheered the Biden administration’s decision to end the immigration rule.

Democrats in both chambers have also grumbled over the decision to jettison $5 billion for global vaccination efforts, which was removed from the funding package after senators could not agree on how to pay for it. Republicans refused to back any new money for responding to the pandemic, so negotiators had to agree on repurposing funds Congress already approved in previous legislation.

Still, some House Democrats signaled they would reluctantly support the package and move to approve the global aid separately, as top White House officials urged Congress to swiftly pass the bill.

“The bill is a start. It should pass immediately, but it’s exactly that — just a start,” said Jeffrey D. Zients, who leads the administration’s coronavirus response. Asked about the Title 42 amendment, he told reporters that “it should remain independent of the urgently needed funding that we talked about today to sustain our Covid response here domestically and our global response.”

“This should not be included on any funding bill,” he added. “The decision should be made by C.D.C., which it has been, and that’s where it belongs.”

Mr. Wyden, the only Democrat to join Republicans in opposing the procedural vote on Tuesday, said he did so because the bill would claw back some pandemic response money that had been set aside for local communities and tribes.

“I cannot support siphoning off essential resources for rural communities, tribes and small businesses in Oregon and nationwide just to pad drug companies’ profits,” Mr. Wyden said in a statement. “It’s pure politics that Republicans insisted vital Covid assistance be drained to pay for this bill.”

Isabella Grullón Paz contributed reporting.