Congressional Republicans on Tuesday struck first on an issue long championed by President Obama, offering legislation to reform the nation's immigration laws just weeks after Hispanic voters bolted from the GOP over the party's harsh stance on immigration.

Sens. Kay Bailey Hutchison of Texas and Jon Kyl of Arizona -- two border-state lawmakers who are retiring -- introduced a bill that would allow young illegal immigrants to stay in the country, though it wouldn't provide them with an expedited path to citizenship. Their proposal was symbolically important for a party that has had trouble winning over the nation's fastest-growing voting bloc, though it's unlikely to get any serious consideration in a lame-duck session dominated by budget issues.

"It's a way to begin the conversation in a calm and reasonable way," Kyl said Tuesday while President Obama was discussing immigration reform with Mexican President-elect Enrique Pena Nieto at the White House.

Some Republicans have called on their party to revamp its secure-the-border message to Hispanics after Obama won roughly 70 percent of the Latino vote despite a stagnant economy and Latinos' anger over a record number of deportations.

"Republicans got the message. To be viable at the national level, they need to get more support from Latino voters," said Alfonso Aguilar, executive director of the Latino Partnership for Conservative Principles. "If you don't get immigration right, Latinos tune you out."

Sen. Marco Rubio, R-Fla., a Cuban-American who has argued against his party's hard-line immigration rhetoric, could lead a conservative reform effort. Rubio's aides said he plans to offer his own proposal to address the children of illegal immigrants in 2013.

"Sen. Rubio has always said the best way is to do it sequentially, starting with the relatively easy stuff," said Rubio spokesman Alex Conant. "What to do with undocumented kids in America is probably the easiest thing. A lot of these problems can't wait; [otherwise] they won't get addressed."

Under the Republican bill introduced Tuesday, young illegal immigrants would receive visas that allow them to enroll in college or enlist in the military. Once they've obtained a degree or served four years, they could then get visas that would allow them to work legally in the United States while they apply for permanent visas.

Some cautioned Republicans against radically changing their immigration positions just because of one election defeat.

"It deflects from other Republican policies that needed a tune-up -- a lot of people perceived the party as too much of Wall Street and not enough of Main Street," said Mark Krikorian, executive director of the Center for Immigration Studies. "There's no way the Republican Party can be the party of amnesty and open borders."

Much is at stake for the president on immigration, too. Obama cited the lack of progress on the issue in his first term as his "biggest failure," but pledged again during his re-election campaign to push for reforms. Mexico's president-elect offered his support Tuesday.

"We fully support your proposal," Pena Nieto told Obama. "We want to contribute. We want to be part of this."