Congressional Republicans, eager to gain ground on an issue critical to their party's future, have begun pushing their own proposals to reform the nation's immigration laws in hopes of bringing Hispanic voters back into the GOP fold.

President Obama carried 70 percent of the Latino vote as he romped over Republican Mitt Romney, and immediately after Election Day two retiring, border-state Republicans introduced legislation that would make the GOP look more welcoming -- or at least less hostile -- to Hispanics.

Sens. Jon Kyl, R-Ariz., and Kay Bailey Hutchison, R-Texas, introduced a bill that would allow illegal immigrants brought to the U.S. as children to earn legal status. It's very similar to Obama's proposed Dream Act, which Republican rejected earlier.

Sen.-elect Jeff Flake, R-Ariz., and Sen. Rand Paul, R-Ky., are expected to introduce legislation of their own next year that would apply a free-market approach to immigration but provide some path to legal status. Paul's office declined to offer specifics.

"Sen. Paul spoke with Sen.-elect Flake recently on a number of issues, including immigration," Paul spokeswoman Moira Bagley. "He looks forward to working with Sen.-elect Flake in the next Congress, and tackling immigration reform and other important issues."

Both measures represent a departure for Republicans who throughout the campaign continued to emphasize the need for tougher laws and a fence along the U.S.-Mexico border and who denounced any pathway to citizenship for illegal immigrants as amnesty for lawbreakers.

"It's amazing how the results of the 2012 election finally opened Republicans' eyes to the changing demographics of this country, particularly in key swing states," said Rodolfo Espino, associate professor in Arizona State University's School of Politics and Global Studies.

Even as Republicans move closer to Democrats on immigration, a compromise that would satisfy both sides may still be a long way off, immigration reform advocates said.

Obama will introduce his own immigration reform proposal in January or February, and people familiar with the president's plan say it will probably mirror a 2007 Democratic bill that would provide a path to citizenship for nearly all of the immigrants now in the country illegally, which some estimates put as high as 20 million people. That goes much further in dealing with illegal immigrants than Republicans have ever been willing to go, but Obama is betting that a newly chastened GOP will be more willing to negotiate.

Republicans have signaled a new willingness to deal on the issue.

"A comprehensive approach is long overdue, and I'm confident that the president, myself, others can find the common ground to take care of this issue once and for all," House Speaker John Boehner, R-Ohio, told ABC News shortly after the election.

But Roy Beck, executive director of NumbersUSA, which seeks to reduce immigration, called the Obama proposal a "Hail Mary pass" that would never pass the Republican-dominated House. Republicans won't gain politically from a compromise with Democrats, Beck said, since new immigrants tend to vote for Democrats.

If a compromise is going to be reached on immigration reform, he said, it will be Democrats who have to give in and abandon efforts to provide a path to citizenship.

"The platform of no compromise won't survive," Beck said.