The White House portrayed it as a step toward compromise.
When President Biden sent his request to Congress last month for aid to Ukraine and Israel, he included a request for more money to help with security at the border with Mexico, a sweetener intended to both address a crisis and win over support of Republicans.
But the move has now left Mr. Biden in a box.
By putting the issue on the table, he ignited demands from the right for broad changes to border policy, leaving his own party divided on a topic that many Democrats see as a political vulnerability heading into 2024 and further complicating prospects for top foreign policy priorities.
The president signaled on Wednesday that he was open to further negotiations with Senate Republicans after they blocked his emergency spending bill.
Mr. Biden now faces a difficult choice about how much to throw himself into talks on an issue that for decades has defied efforts to reach bipartisan compromise. And he will have to decide how far to go in giving in to conservative demands that he substantially choke off the number of migrants admitted to the United States while their asylum claims are considered.
“The president of the United States should be involved,” said Senator Lindsey Graham, Republican of South Carolina. “Everybody behind me will vote to aid Ukraine if we can get the border right.”
The White House has limited time to reach a compromise with a Congress whose members are set to leave for recess next week. Senator Christopher S. Murphy, Democrat of Connecticut, told reporters he was prepared to continue talks through the weekend.
On Thursday, however, Senate Republicans showed no sign of backing down.
Senior Biden administration officials have told both sides in the negotiations that the White House is open to making it more difficult to gain asylum in the United States, according to four people familiar with the matter. One such measure would impose a stricter definition for migrants to meet when they claim they need refuge because they fear persecution in their home countries.
But Republicans say that is not enough. They want the United States to impose policies that would make most migrants ineligible for asylum and require them to wait in Mexico until their case is processed.
The impasse has left the heart of Mr. Biden’s foreign policy — support for the war in Ukraine — hanging in the balance.
“We’re obviously deeply concerned about this. We’ve got a few more weeks here,” John Kirby, a White House spokesman, said of securing aid for Ukraine. “There are a small number of Republicans that want to hold that aid hostage for some pretty extreme border policies that the president is not willing to talk about. That said, he did say we’re willing to negotiate in good faith.”
On Wednesday, the president implored Congress to put aside “petty, partisan, angry politics” and pass the $111 billion bill. He said failure to do so could enable President Vladimir V. Putin of Russia to reclaim momentum in the war.
President Volodymyr Zelensky of Ukraine has said he needs the U.S. funding to defend against Russia’s invasion. Mr. Kirby said Thursday that the White House was “not in a position to make that promise to Ukraine, given where things are on the Hill.”
Republicans, meanwhile, are calling attention to migration at the southwest border, which has topped 8,000 crossings a day in recent days — and become a major political liability for Mr. Biden.
Immigration advocates say the impasse is evidence that Mr. Biden never should have paired the wartime funding with immigration reform in the first place.
“Grouping the issues together in the supplemental funding request strategically and substantially was a catastrophic error because it was the signal, it was the beginning of what came to fruition yesterday when the president said immigrant communities are a bargaining chip,” said Heidi Altman, the policy director at the liberal National Immigrant Justice Center. “That’s a betrayal.”
Mr. Kirby said the White House had no regrets about grouping the administration’s foreign policy priorities with immigration in the bill because they are all of an “urgent nature.”
“It was in our national security urgent request: $6 billion for the border. We share a sense of urgency, so act on that,” Mr. Kirby said.
Mr. Biden’s aides have consistently responded to Republican attacks on the subject of the border by pointing to a plan proposed at the start of the Biden administration that would have established a pathway to citizenship while increasing funding for the border. Republicans, they say, would rather use the migration crisis as a political weapon than compromise on solutions.
The White House has received backlash from both sides, highlighting the challenge of reaching a compromise over one of the most polarizing issues in domestic politics.
Senator Alex Padilla, Democrat of California, said in a statement co-signed by 10 other Senate Democrats that “using a one-time spending package to enact these unrelated permanent policy changes sets a dangerous precedent and risks assistance to our international partners.”
Senator John Cornyn, Republican of Texas, spoke for his G.O.P. colleagues on Thursday when he boasted about pushing immigration to the forefront of the national conversation. “It looks like we’ve got the president’s attention,” he said.
Peter Baker, Eileen Sullivan, Karoun Demirjian and Hamed Aleaziz contributed reporting.
