Two key architects of the Senate immigration bill expressed confidence that the bill will make it through the House, despite opposition from conservative lawmakers.
McCain said he was "hopeful that we can" pass a bill in the House, citing a wide coalition of support for reform, but acknowledged there is much work to be done.
"I believe that Speaker Boehner [R-Ohio] has a tough job ahead. I admire his leadership, but we've got a lot of work to do," he said.
The Senate passed the bipartisan bill drafted by the Gang of Eight earlier this week on a 68-32 vote, with the support of every Democrat and 14 Republicans.
Conservatives in the House have expressed opposition to the bill, however, which they say doesn't do enough to improve border security. They also oppose a pathway to citizenship, and some Republicans have suggested a piecemeal approach in the House is the only way to pass reform.
Schumer, however, said he expects a full reform package to pass the House before the end of the year. He also cited the coalition of support, and said the movement to pass reform "has the potential to be one of the greatest civil rights movements we've ever seen."
But he also noted the electoral implications of failure.
"The national Republican leadership will tell John Boehner, if you don't pass a bill, we're going to be the minority party for a generation. And he's not only the House leader, he's a party leader," Schumer said.
The Republican Party has made an effort to broaden its appeal to minorities, particularly Hispanics, after a disappointing showing in the 2012 election. The GOP lost Hispanics by more than 40 points on the presidential level.
McCain acknowledged that while passage is "not going to be easy," the electoral implications will spur action. But he stopped short of urging Boehner to bring a bill to the floor without the support of the majority of his caucus in the House.
"I really hesitate to tell Speaker Boehner exactly how he should do this, but I think Republicans realize the implications for the future of the Republican Party in America if we don't get this issue behind us," he said.