A bipartisan group of senators has agreed on a compromise to crack down on the surge of migrants across the United States border with Mexico, including reducing the number who are allowed to live and work in the country temporarily, but a final deal depends on resolving critical funding disputes.

Much of the recent haggling over the emerging agreement — and a point of contention for its critics — has been about how to limit the number of people who are granted parole, a status that allows migrants without visas to live and work in the United States temporarily. Drawing broader support for the plan in the Senate would clear away the biggest obstacle to congressional approval of tens of billions of dollars in aid to Ukraine for its war against Russian aggression, which Republicans have said they will block absent a deal to clamp down at the border.

The G.O.P. considers parole a dangerous loophole that fuels illegal immigration and must be tightly closed. To many Democrats, it is a crucial tool that allows the administration to treat desperate migrants humanely, which must be preserved, particularly for vulnerable populations fleeing failing states and war.

Bridging the gap will likely hinge on the two sides agreeing on how much federal money to spend to try to decrease net immigration numbers, and persuading Republicans that the deal’s measures will be effective. On Monday, Senate leaders warned that getting a final deal depended on resolving those remaining differences.

“It’s certainly not a done deal yet; there are a handful of issues that have not yet been agreed to,” Senator Chuck Schumer, Democrat of New York and the majority leader, said on the floor. “On something as complicated as the border, it’s not just what we do that matters. It’s how we do it.”

Here’s how parole works, and why it’s central to sealing any border deal in Congress.

Parole authority, which has existed since the 1950s, allows the government to extend migrants a special status to remain in the United States for a certain period of time. It was designed to be used only in cases of humanitarian need, or if there was a public benefit to allowing a migrant into the country.

But administrations have interpreted that guidance in different ways, sometimes ushering in whole groups of migrants under the authority.

The Biden administration also paroled in roughly a million migrants under a number of programs geared toward helping specific vulnerable groups, such as people fleeing the war in Ukraine and the Taliban government in Afghanistan, as well as migrants from economically devastated countries like Haiti, Venezuela, Cuba and Nicaragua who have sponsors already in the United States. Republicans have sought to limit nearly all of those programs.

According to data compiled by the Transactional Records Access Clearinghouse at Syracuse University, the use of parole authority to allow otherwise inadmissible migrants into the United States has also ballooned under the Biden administration, though the percentage of those admitted is roughly comparable to intake figures seen under President Barack Obama and during the early part of the Trump administration.

Republicans have railed against what they call the practice of “catch and release,” in which migrants are briefly detained upon their arrival into the United States but then quickly granted parole and released to await immigration court hearings.

The emerging Senate deal seeks to reduce parole numbers by tightening immigration enforcement and speeding up processing. It would make it harder for migrants to claim asylum, expand detention capacity in the United States and expedite the expulsion of migrants who lack lawful reasons to stay in the country.

The plan would also add staff to the Border Patrol and asylum officer corps tasked with processing migrants through the immigration intake and court systems. And it includes a fail-safe mechanism that would shut down migrant intake altogether if the system became overwhelmed, a level that officials put at more than 5,000 daily encounters with people trying to cross the border. In November, daily encounters at the southwest border averaged above 8,000 per day, according to statistics published by U.S. Customs and Border Protection, and last month, encounters peaked at over 12,000 per day, according to reports.

Proponents contend that those measures would effectively reduce the number of migrants who would need to be granted parole. But Republicans have sought a hard cap on the number of migrants who can receive parole, as well as the elimination of group-based parole. The deal does not include either.

The compromise under discussion turns on an elusive agreement on funding.

Negotiators believe that if they secure substantial money for the beefed-up enforcement and processing powers they have agreed upon, parole numbers will decline without the hard caps that Republicans have demanded.

But lawmakers are still bargaining over how much money must be included to do that job. If they cannot reach agreement, the compromise could collapse.

House Republicans, including Speaker Mike Johnson, have threatened to block any deal that does not eviscerate most group-based parole programs and reinstitute a policy of keeping migrants who cannot be held in detention facilities on the Mexican side of the border. In recent weeks, Republicans in both chambers have also rallied around demands for strict numerical annual caps for parole, calling that a “red line” to ensure that the president cannot abuse his parole authority.

“None of us trust the Biden administration to implement the law,” Senator Lindsey Graham, Republican of South Carolina, told reporters last week. “So there needs to be a hard cap on parole.” Mr. Graham is not a participant in the bipartisan negotiations, but his support is considered critical for rallying other Republicans around an eventual bill.

For the most part, Democrats regard parole as an important tool for managing an often chaotic situation at the border and the limitations of U.S. detention facilities and immigration court backlogs. They also fear what might happen to vulnerable populations like Afghans and Ukrainians if the president’s ability to offer parole to groups fleeing acute humanitarian crises were scaled back, or capped at a low threshold.

While many Democrats have rallied around the idea of reducing the influx of migrants unlawfully trying to enter the United States, they have balked at the Republican demands to directly limit parole authority.

And some progressive Democrats say any deal that tries to reduce parole numbers by tightening restrictions on migrants, including shutting down intake if the number of border crossings exceeds a certain level, is unacceptable.

Hamed Aleaziz and Zolan Kanno-Youngs contributed to this report.

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