Rep. Bob Goodlatte, R-Va., the chairman of the House Judiciary Committee, rejected a "special pathway to citizenship" to immigrants living in the country illegally, but indicated that he would be receptive to a path to legalization.
Outside groups recognize the political headwinds and are simply pressing for a vote.
"There's a lot of different ways they can get to an outcome on immigration reform and the fact is we can't pretend to be able to control the process. But what we can say is that we want a vote. Just vote on something," said Janet Murguia, president of National Council of La Raza.
"We know that if there is a bill that is voted on in the House of Representatives it will be conferenced with the Senate bill, so just vote on something and let the conference be the place where we can negotiate the differences. ...But for us, give us a vote. We deserve a vote."
Democrats cast the issue as a moment in history for Boehner, with nothing less than the future of the Republican Party at stake in national elections. The Hispanic population in the United States is up 65 percent since 2000, with millions of new voters who delivered a shellacking to the GOP in last year's presidential election.
GOP Nominee Mitt Romney's defeat was blamed in part on his 27 percent of the Hispanic vote.
"It's an existential dilemma for the Republicans," said Rep. Peter Welch, D-Vt. "The folks in gerrymandered (House) districts can just say no, but if they want to have any national future, the adults in the room have to say yes."
Rep. Gerry Connolly, D-Va., motions toward the portraits of past speakers lining the hallway outside the House chamber and asks, "What's the point of having the job and the title if you're not going to do something with it?"
"Immigration reform is one of the top three or four issues in the country and will affect the next generation of Americans and it can't continue to be ignored," he said.
If Congress is unable to produce immigration legislation, Democrats are certain to blame the GOP and look to capitalize politically, not only with Hispanics but independent voters frustrated with the gridlock.
Asked if failure was a winner for Democrats, Connolly said, "We all presume so. In the short run that's less clear than in the long run."
Associated Press writers Erica Werner contributed to this report.
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