The soaring number of people crossing into the United States from Mexico has been a political vulnerability for President Biden for the past three years, chipping away at his approval rating and opening him up to political attacks.

But now, the crisis is threatening to upend America’s support for the war in Ukraine, throwing the centerpiece of Mr. Biden’s foreign policy into jeopardy.

After a meeting with Mr. Biden at the White House on Wednesday, Speaker Mike Johnson insisted that the Republican-led House would not pass legislation to send aid to Ukraine unless Democrats agreed to sweeping new restrictions at the U.S.-Mexico border.

And even if the two sides do come to some sort of agreement, many Republicans, especially in the House, would be loath to give an election-year win to Mr. Biden on an issue that has given them a powerful line of criticism toward the White House. The issue is also at the center of the candidacy of Mr. Biden’s likely opponent this fall, former President Donald J. Trump.

The stalemate shows how the debate over immigration in the United States is no longer just about the border. The issue is spilling over into other parts of Mr. Biden’s agenda, taking on outsize influence as Republicans use it to block the president’s top foreign policy priorities.

“I think the vast majority of members of Congress support aid to Ukraine,” Mr. Biden told reporters on Thursday before traveling to Raleigh, N.C. “The question is whether or not a small minority are going to hold it up, which would be a disaster.”

Mr. Biden has characterized aid to Ukraine as a matter of American leadership on the global stage. If the United States fails to send more, he warned last month, other allies may back off their own commitments. President Vladimir V. Putin of Russia, he said, could reclaim momentum in the war.

But House Republicans have so far been unmoved. Mr. Johnson said this week he was focused on “national security and a humanitarian catastrophe” within U.S. borders.

Janet Napolitano, former governor of Arizona and homeland security secretary in the Obama administration, said she could not remember a time in recent decades when so much of an administration’s agenda hinged on immigration policy.

The legislation that Mr. Biden is pushing does not only include Ukraine aid. It also has money for Israel and Taiwan — and billions of dollars to bolster security along the U.S.-Mexico border, just not enough to satisfy Republican demands.

“They looked at this as an opportunity and they’ve taken it,” said Ms. Napolitano, who described the politics of immigration right now as “dire.”

“It really means the president has got to go as far as he can and to work with those across the aisle to get a package through,” she said.

Mr. Biden has said he is willing to make compromises on the border. Democratic negotiators, with the approval of the White House, have signaled that they would consider proposals making it more difficult to gain asylum in the United States.

The White House has appeared less willing to substantially restrict humanitarian parole, a program that has allowed thousands of Afghans, Ukrainians and migrants at the border to enter the United States.

While members of Congress are still debating details of that policy, it’s not clear that compromise is in the cards.

In an interview with Fox News on Wednesday evening, Mr. Johnson made a point of saying he had spoken about the negotiations with Mr. Trump, who encouraged him to oppose compromising.

Mr. Biden is also facing pushback from progressives, who do not want to see restrictions on asylum.

“Republicans are holding foreign aid hostage to extract extreme immigration measures that would not solve the problem,” said Nanette Barragán, Democrat of California and chair of the Congressional Hispanic Caucus. “Many of us support aid to Ukraine, but not at the expense of these extreme immigration policies on the table.”

Jim Kessler, executive vice president for policy at Third Way, a center-left think tank, said the linking of aid for Ukraine to border restrictions was “unprecedented.”

“It’s hard to imagine a time previously in our nation’s history where something that is so important from a national security standpoint, that would normally unite all Americans of both political parties, is caught up in games,” Mr. Kessler said.

The White House initially portrayed the decision to pair immigration with the military funding request as an enticement, or at the very least an attempt at compromise, to win over Republicans who had been calling on Mr. Biden to get tougher on the border.

William B. Taylor Jr., who served as ambassador to Ukraine from 2006 to 2009, said there would be severe consequences both for Ukraine and the United States if the strategy ended up failing.

“I imagine that the calculation was that there’s a lot of support for these and if we put them all together we’ll have a winning strategy,” Mr. Taylor said. But if the White House cannot reach a deal, he added, it would undermine “a crucial component of U.S. foreign policy.”

“That U.S. leadership would be badly damaged if we can’t provide the assistance to the Ukrainians to allow them to stop the Russians now,” he said. “It has enormous implications.”