Although House Republican leaders say they want to solve the issue, which has become a political drag for the GOP, many rank-and-file House Republicans have shown little inclination to deal with it. With just a few legislative weeks left in the House, it's unclear whether lawmakers will vote on any measure before the year is out.
Among the exceptions are Republican Reps. Mario Diaz-Balart of Florida and Darrell Issa of California, who have been working on possible legislation.
Diaz-Balart has said his bill would help those immigrants here illegally to "get right with the law," purposely avoiding the word legalization that he said is interpreted differently in the fierce debate over immigration. Diaz-Balart is slated to meet with the president on Tuesday.
The congressman mentioned the session in an interview with Florida radio station WGCU and his office confirmed the meeting.
Determined to rally support, outside groups such as the U.S. Chamber of Commerce and Bibles, Badges & Business for Immigration Reform are descending on the Capitol Tuesday to lobby lawmakers to vote this year on immigration legislation.
Randy Johnson, senior vice president of the Chamber, told reporters on a conference call Monday that the effort is "about moving votes on the Hill in the right direction."
Johnson said he was hopeful that the House could pass one or two of the single-issue bills before the end of the year, and left open the possibility of action early next year — an election year.
"I don't think it's the end of world if we can't get it done by early February," Johnson said. He said if it drags on until April or May, the prospects are dim.
Separately, Rep. Jeff Denham of California became the first Republican to back the House Democratic bill. Denham represents a swing district in northern California northeast of San Jose. He won his seat in 2012 with 53 percent of the vote.