Hoping to boost his party’s image with Hispanic voters, Sen. Marco Rubio (R-Fla.) launched a one-man push Thursday to promote a modified version of legislation to benefit undocumented children whose parents brought them to this nation.

Sen. Marco Rubio (R-Fla.). (Haraz N. Ghanbari/AP)

Rubio’s working draft of similar legislation would grant legal immigration status to such children who were brought to the United States illegally by their parents. It would require that they graduated from college or served honorably in the military.

His legislation would grant such a young adult a visa to stay in the country but would require them to apply for citizenship just like other immigrants, as opposed to the system in a bill drafted by Sen. Richard Durbin (D-Ill.), which would give an advantage to those immigrants in gaining citizenship.

“The issue of kids who were brought to this country and are here in an undocumented status through no fault of their own, who are high achievers and have much to offer us in the future, I think there’s broad bipartisan support for the notion that those kids are in a different category than the vast majority of people who find themselves in this country undocumented,” Rubio told a group of reporters in his conference room.

Rubio acknowledged that his legislation was a work in progress — he has to figure out how long the visa would last for these immigrants, and what happens to their parents is still to be determined. “I don’t have the magic solution,” he said. Instead, Rubio said he wants to work with senators of both parties to come up with a bipartisan solution that has so far evaded Congress on the contentious issue.

The last vote on Durbin’s Dream Act, in December 2010, received 55 votes, a handful shy of the 60 needed to end debate and move toward a final vote. The last true debate on immigration — a comprehensive reform package negotiated by a bipartisan group of more than 20 senators — fell apart in June 2007, ending with 46 votes in favor.

The 2007 effort ended with many GOP senators walking away from the legislation they had been negotiating, including Senate Minority Leader Mitch McConnell (Ky.) and Sen. John Cornyn (Tex.), who has been chairman of the National Republican Senatorial Committee since 2007. On the 2010 Dream Act vote, every member of the Republican leadership opposed the measure.

Democrats have dismissed Rubio’s effort as insufficient for children who go on to serve in the nation’s military or go to its best universities, ridiculing it as an attempt for Republicans to get political cover.

“The DREAM Act is there for a purpose: to give young men and women, who came here when they were really just kids, an opportunity to join the military or to go to school. And I am not going to agree to anything that is short of allowing these young men and women to become citizens after they make the sacrifices they do,” Senate Majority Leader Harry M. Reid (D-Nev.) said Tuesday when asked about Rubio’s effort.

Rubio, the son of Cuban immigrants, has become the leading Republican pushing to soften his party’s image on issues key to Latino voters, who could play a key role in the 2012 elections in several battleground states. However, he said he has not sought the support of the likely GOP nominee, Mitt Romney, or GOP leaders for his legislation; that would come later, he said.

Earlier Thursday, appearing at a policy breakfast sponsored by National Journal, he pushed many of the same issues but once slipped up and referred to his tenure as “vice president” rather than senator. That sparked even more questions about his interest in being Romney’s running mate, speculation that he has tried to tamp down ever since he arrived as the only Latino Republican in the Senate.

Tuesday afternoon, he again said he had no interest in becoming vice president, but did hint that he has broader ambition that could lead to a run for national office down the road.

“I am really committed to doing a good job in the Senate,” the Florida Republican told a dozen-plus reporters Thursday. “If I do a good job in the Senate, three, four, five, six years from now, I’ll have a different opportunity ... to do things inside of government and outside of government.”

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