Even ardent proponents of an immigration-law overhaul are, at best, cautiously optimistic. In June, a broad immigration overhaul — with a 13-year path to citizenship for the 11 million immigrants now in the country illegally, and stricter border security provisions that would have to be in place before the immigrants could gain legal status — passed the Senate with bipartisan support. But that legislation has largely stalled in the Republican-controlled House, where Mr. Boehner has rejected any negotiations with the Senate over its comprehensive bill.

“This is obviously a long, hard road,” said Senator Charles E. Schumer of New York, the No. 3 Democrat, who helped negotiate the Senate bill, “but I think since August, the number on the other side vehemently opposed has stayed the same, the number who think it should go forward has grown, and numbers in the wide middle are less opposed than they used to be. But that doesn’t guarantee an outcome one way or another.”

Republican Party leaders, backed strongly by business groups, have said an overhaul is critical if they are to repair their political position with Latino and other immigrant voters.

Barry Jackson, Mr. Boehner’s former chief of staff, is consulting for the U.S. Chamber of Commerce, which supports an overhaul that expands high-technology visas and guest worker programs.

But immigration is less of an issue during midterm elections, when immigrants are not as likely to vote and House members in safe districts are insulated somewhat from the wrath of more moderate swing voters. Often the biggest threats to Republicans are primary challenges from more conservative candidates who say that changing the immigration status of someone who is in the country illegally amounts to amnesty for a lawbreaker.

Image William Kristol, a conservative editor, opposes a Republican push on immigration. Credit... Chip Somodevilla/Getty Images

Representative John Carter, Republican of Texas and one of the Gang of Eight House members who tried to forge a bipartisan overhaul, was quoted by the Capitol Hill newspaper Roll Call as saying that an election year is not the time to press forward. “Immigration is a very, very contentious issue,” he said.