Nevertheless, only 14 of the Senate's 46 Republicans voted for the bill and many House Republicans complain that the 11 million illegal residents would be mainstreamed into American society before the border is fully secured.

Last November's presidential election, in which Obama captured more than 70 percent of the growing Hispanic vote, was a wake-up call to Republicans that their party must do more to appeal to minorities.

Former President George W. Bush, who failed to win passage of a comprehensive immigration bill when he was in office, on Wednesday said that he hoped there would be a "positive resolution" to Congress' immigration debate

Speaking in Dallas at a naturalization ceremony, the two-term Republican president said, "We have a problem. The laws governing the immigration system aren't working. ... The system is broken."

The call for comprehensive reform resonates with some Republican senators, who have to run in statewide elections, and with some prospective Republican presidential candidates.

But it holds less appeal to House Republicans, many of who fear conservative Tea Party challenges if they back a pathway to citizenship for the 11 million, a core demand of Obama and his fellow Democrats.

According to a recent study by the nonpartisan Cook Political Report, which tracks congressional races, only 24 of the 234 House Republicans represent districts that are more than 25 percent Hispanic.

David Wasserman, who conducted the Cook study, said most House Republicans believe they could defeat a Democratic challenger in the general election.

But "they don't know if they will face a Republican primary challenge if they vote for an immigration bill backed by the president," Wasserman said.

For many House Republicans, support for a comprehensive bill with the pathway to citizenship is tepid at best.

Passing such legislation is "not urgent," said Representative James Lankford of Oklahoma, a member of the House Republican leadership team.

"If we run out of time at the end of the year, I don't think we push it. This is a problem that has festered for decades," he added.