This week in the Republican-led House, G.O.P. representatives plan to kick off their formal effort to impeach Alejandro N. Mayorkas, the homeland security secretary who has been one of their favorite punching bags.

Over in the Senate, a group of Republicans is pressing to finalize and announce an emerging deal they have negotiated with Mr. Mayorkas in a series of high-stakes talks about how to address the surge in migration at the U.S. border with Mexico.

The disconnect highlights a broader split among Republicans that could complicate Congress’s efforts to reach an agreement on new border policies — and further imperil the push to fund the government and send additional aid to Ukraine in the coming weeks. While the House G.O.P. has pushed a hard line on immigration, insisting on severe restrictions and the ouster of the Biden administration official who oversees it, some Senate Republicans have instead sought to force a compromise on the issue, relying in part on Mr. Mayorkas to seal the deal.

For more than a year, Mr. Mayorkas has been a prime target of congressional Republicans, especially in the House, who have accused him of intentionally flouting the law to allow migrants and illicit drugs into the country. But in recent weeks, Mr. Mayorkas has emerged as a key player in the bipartisan Senate talks, which aim to satisfy G.O.P. demands that a border crackdown be part of any deal to speed security assistance to Ukraine.

Mr. Mayorkas has been the Biden administration’s chief envoy to those negotiations, huddling with senators on a near-daily basis to cement a deal that Senate leaders hope to vote on this month. It is an awkward position for a person who is about to be charged with high crimes and misdemeanors on the other side of the Capitol.

The House Homeland Security Committee is planning to ready articles of impeachment against Mr. Mayorkas in a matter of weeks, according to its chairman, Representative Mark E. Green, Republican of Tennessee. Mr. Green has said his panel will hold only three or four hearings in January, beginning with a panel of state attorneys general on Wednesday. That means Republicans in both chambers will soon be forced to decide whether they can back a deal brokered by a cabinet secretary whom their party is also trying to remove from office.

“It raises a huge question about whether some Republicans, when this deal is announced, are going to accept a deal negotiated by Mayorkas,” said Andrea Flores, the vice president for immigration and campaigns at FWD.us. She argued that the contrast “really suggests a lack of support from House Republicans” for the Senate efforts to strike a border security deal.

“I don’t understand how they take a Mayorkas-branded deal,” she added.

While Speaker Mike Johnson has not condemned the Senate negotiations, he has long maintained that he would support only a deal that closely mirrored the terms of a border security bill the House passed last year, which sought to reinstitute and codify a number of Trump-era enforcement, detention and deportation programs. In an interview on CBS News’s “Face the Nation” on Sunday, he went a step further, accusing Mr. Mayorkas of being untrustworthy as he defended the House’s decision to move forward with its impeachment efforts, even if it complicated efforts to strike a border bargain.

“Secretary Mayorkas is not a good-faith negotiating partner,” Mr. Johnson said. “He is unwilling to enforce existing federal law. Why would we believe that he would do any new provision?”

Mr. Green also questioned Mr. Mayorkas’s motives, saying in a statement on Monday that “Secretary Mayorkas trying to influence a border security package while consistently refusing to enforce the laws that would actually secure the border is a bit like the arsonist claiming he’s trying to help put out the fire.”

That sentiment has left Senate Republicans scrambling to play down Mr. Mayorkas’s role in the border talks.

“He is not actually doing this policy,” Senator James Lankford of Oklahoma, the lead Republican participant in the talks, said on “Fox News Sunday.” “He is actually providing what’s called technical assistance.” Mr. Lankford suggested that the secretary’s role has been limited to answering operational questions about how the changes lawmakers are contemplating might actually play out on the ground.

Mr. Mayorkas, for his part, has largely brushed off the accelerating efforts to impeach him. In a “PBS NewsHour” interview earlier this month, he said he was focused instead on “achieving the missions, delivering solutions, working with Republican and Democratic senators alike to fix a fundamentally broken immigration system.”

The emerging border deal still faces several practical and political hurdles. While negotiators have agreed on changes that would make it more difficult for migrants to claim asylum and expand the administration’s ability to deport unlawful arrivals swiftly, they have yet to come to terms on whether and how to limit parole. Under that program, certain categories of migrants are allowed to live and work in the United States while awaiting their day in immigration court.

Republican leaders are under enormous pressure from the party’s right wing not to accept a deal that fails to eliminate parole for most categories of migrants. In the Senate, leaders are expected to attempt to address hard-line Republicans’ concerns during a special meeting on Wednesday.

In the House, however, right-wing Republicans are threatening to block not only efforts to approve military aid for Ukraine, but also any bill to fund the government unless their border demands are satisfied. That has put Mr. Johnson, who has said he does not want to shut down the government over a border security dispute, in the uncomfortable position of trying to find other ways to please his impatient base.

Mr. Mayorkas’s impeachment, though long considered a nonstarter because of internal House G.O.P. resistance, has recently become a promising rallying point for the conference. In the fall, eight Republicans helped block efforts led by Representative Marjorie Taylor Greene, Republican of Georgia, to hold a snap impeachment vote against Mr. Mayorkas on the House floor.

But leaders believe that in the weeks since, many of those holdouts have fallen into line, placated by the fact that the House Homeland Security Committee is giving the charges formal consideration.

The hardening of anti-Mayorkas sentiment in the House has left some Republican senators who favor a border deal to make halfhearted defenses of the homeland security secretary.

“He’s working for the president of the United States, and these are the president’s policies,” Mr. Lankford said of Mr. Mayorkas on Fox. “The problem is President Biden’s policies on the border, not Mayorkas, in the way he’s carrying it out.”

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