WASHINGTON — More than six in 10 Americans now favor allowing illegal immigrants to eventually become U.S. citizens, a major increase in support driven by a turnaround in Republicans’ opinions after the 2012 elections.

The finding, in a new Associated Press-GfK poll, comes as the Republican Party seeks to increase its meager support among Latino voters, who turned out in large numbers to help-re-elect President Barack Obama in November.

Emboldened by the overwhelming Hispanic backing and by shifting attitudes on immigration, Obama has made overhauling laws about who can legally live in the U.S. a centerpiece of his second-term agenda.

In the coming weeks, he’s expected to aggressively push for ways to create an eventual pathway to citizenship for the estimated 11 million illegal immigrants already in this country.

The poll results suggest the public overall, not just Hispanics, will back his efforts. Sixty-two percent of Americans now favor providing a way for illegal immigrants in the U.S. to become citizens, an increase from 50 percent in the summer of 2010, the last time the AP polled on the question.

In a 2009 poll, some 47 percent supported a pathway to citizenship for illegal immigrants.

Further boosting the president on the issue, Democrats have opened a 41 percent to 34 percent advantage as the party more trusted to handle immigration, the first time they’ve held a significant edge on the matter in AP-GfK polling. In October 2010, Republicans held a slight edge over Democrats, 46 percent to 41 percent, on the question of who was more trusted on immigration.

Much of the increase in support for a path to eventual citizenship has come among Republicans. A majority in the GOP — 53 percent — now favor the change. That’s up a striking 22 percentage points from 2010. Seventy-two percent of Democrats and 55 percent of independents like the idea, similar to 2010.

The findings suggest those GOP lawmakers weighing support for eventual legal status for illegal immigrants could be rewarded politically not just by Democrats and independents, but also by some in their own party as well. This comes amid soul-searching in the party about how the GOP can broaden its support with Latinos, who backed Obama over Republican presidential nominee Mitt Romney 71 percent to 27 percent in November. Romney received less support from Latinos than Republican President George W. Bush did. But his slice was on par with candidates Bob Dole in 1996 and George H.W. Bush in 1992.

Some Republicans have concluded that backing comprehensive immigration reform with a pathway to citizenship is becoming a political necessity. Many lawmakers remain strongly opposed, and it’s far from clear whether Congress will ultimately sign off on such an approach. But in the Senate, a bipartisan group of lawmakers is working to draft immigration legislation, and Sen. Marco Rubio, R-Fla., a possible 2016 presidential candidate, has offered proposals that would ultimately allow illegal immigrants to attain legal status.

One poll participant, Nick Nanos, 66, of Bellmore, N.Y., said that providing a way for illegal immigrants to become citizens would respect America’s history as a nation built by immigrants.

“We act as if our grandparents got here legally. Don’t want to ask a single Indian about that,” Nanos said in a follow-up interview. “I don’t think that most of us can solidly come to a point where our grandparents or great-grandparents or great-great-grandparents were here legally. What does that even mean?”

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