A crush of asylum claims at the U.S.-Mexico border is complicating an already intractable immigration debate on Capitol Hill, pulling the two parties further apart and threatening to undermine what some lawmakers have viewed as the best hope in a decade for Congress to forge a comprehensive immigration deal.

For decades, bipartisan discussions on such a compromise focused on pairing beefed-up border security with a pathway to legalization for undocumented immigrants and expanded legal pathways to entry. But in recent years, an explosion in the number of migrants asking for asylum — a protected status for those fearing persecution in their home country — has scrambled the equation, exposing deep political and moral divisions.

The shift helps explain why talks on Capitol Hill to find a consensus on a comprehensive immigration overhaul have sputtered, despite lawmakers’ hope that the expiration this week of Title 42 — a pandemic-era policy that had let authorities swiftly expel migrants — would force Congress to act.

“It’s changed,” Senator Chris Coons, Democrat of Delaware, said of the immigration overhaul debate. “We managed to get to an agreement that put a significant amount of money into border security last time. Making similar progress on asylum is very, very different, because it’s a values issue.”

Mr. Coons is one of a group of about eight Republican and Democratic senators who have been privately talking for months about an immigration compromise, but have splintered in recent days over how to handle asylum.

Asylum claims were intended to be reserved for people fearing persecution on the basis of their race, religion, nationality, political opinion or membership in a social group to seek protection on U.S. soil. But in recent years, they have increasingly become a go-to tactic for migrants with no other options to enter the United States, and who know it could take years before their cases are heard and — if unfounded — rejected.

Pending asylum claims before the immigration courts have grown more than sevenfold over the last decade, according to data analyzed by the Transactional Records Access Clearinghouse at Syracuse University, forcing the issue into the center of the congressional debate.

In recent months, Republicans and Democrats have embraced radically different positions on how to address abuses of the asylum system.

Republicans have proposed steps to restrict access across the board, pushing legislation through the House this week that would require migrants claiming a credible fear of persecution to wait outside the United States for their cases to be heard in court. They only narrowly stopped short of approving language that would have shut down the asylum system if the United States ran out of detention beds.

Democrats have largely gone in the other direction, embracing a right to seek asylum protections as intrinsic to the character of the United States and calling for expanding other pathways to legal immigration to alleviate the strain.

Bipartisan talks in the Senate about an immigration overhaul hit a snag recently over the most high-profile proposal to emerge from the discussions: a proposal by Senators Kyrsten Sinema, independent of Arizona, and Thom Tillis, Republican of North Carolina, to give the Biden administration a two-year authority to speedily expel migrants trying to cross the border, but with a carveout for asylum claims that did not exist under Title 42.

None of the Democrats from the group have supported the legislation, while some of the Republicans also backed away, claiming the bill did not do enough to dissuade fraudulent asylum claims.

Ms. Sinema and Mr. Tillis have said that they saw their proposal — which has earned some support from Republicans and moderate Democrats in the Senate and House — as a potential centerpiece for a broader bill that would include border security measures and a pathway to citizenship for unauthorized migrants brought to the United States as children.

During the last Congress, they attempted to rally support around a similar blueprint, but ran out of time to complete their work. Mr. Tillis and others in bipartisan group estimated that they were still weeks away from presenting anything resembling a comprehensive proposal.

Experts said the lag could further complicate efforts to strike a deal, pointing out that the situation at the border, particularly around the asylum issue, is changing faster than lawmakers drafting bills are responding to it.

“At the end of the last Congress, they actually had the most balanced, most bipartisan support in the outline that they circulated,” said Jennie Murray, the president and chief executive of the National Immigration Forum, a pro-immigration group. She noted that at the time, what seemed like proactive provisions — such as a bid to curb asylum claims by extending the expulsion authority of Title 42 — would now be seen by some Democrats as an unacceptable retrenchment.

“We’re just in a completely different context,” she added.

As the situation at the border changes, the parties are moving further into opposite corners. This week, the administration rolled out a new set of rules to address an expected spike in asylum claims after Title 42’s demise, despite calls from several leading Democrats, including Representative Jerrold Nadler of New York, the top Democrat on the House Judiciary Committee, to reconsider.

Mr. Nadler responded by joining the rest of the Democrats in New York’s congressional delegation, including Representative Hakeem Jeffries, the minority leader, and Senator Chuck Schumer, the majority leader, in appealing to President Biden in a letter “to expand the issuance of parole to asylum seekers,” by eliminating a 150-day waiting period for applicants to be allowed to work.

Republicans, by contrast, have accused the administration of holding up progress in Congress by refusing to take more restrictive steps at the border.

“My advice to the administration is to change your asylum policies,” said Senator Lindsey Graham, Republican of South Carolina, a key negotiator in past efforts to write comprehensive immigration bills. “Stop the flow. If you can control the flow, regain control of the border, then you have a chance to talk about what we’ve been talking about for 15 years. Without that, you’re probably going nowhere.”

Few see an easy way out of the impasse, especially as Washington gears up for a presidential election that could pit Mr. Biden against former President Donald J. Trump, whose restrictive immigration policies many Republicans are trying to revive.

“This is about the presidential election in 2024,” said Representative Lou Correa, Democrat of California. “I think it’s going to be very difficult under Biden’s first term to actually move in that direction.”