“It’s not that we pass this bill and all of the sudden the Republican brand with minorities and ethnic communities is ‘Hey, great!’ ” Diaz-Balart concedes. “But if we pass this bill, then Republicans can start going to those communities.”

The question, however, is: Going to them with what? In addition to fixing a broken system, Republicans say that a reform law would give them a credibility card with Hispanic voters, allowing them to talk about the conservatism they believe many Latinos naturally subscribe to.

These voters have had “immigration earmuffs,” said Jennifer Korn, executive director of the conservative Hispanic Leadership Network. They’re not listening to Republicans on the economy, education, the deficit and other issues because immigration hasn’t been addressed yet, she said. Once that issue is resolved, Republicans can emphasize concerns that are less fraught with emotion and ethnic politics: smaller government, erasing the deficit, tax reform and pulling Obamacare “out by its roots,” in the words of Senate Republican Leader Mitch McConnell.

But what if a majority of Latinos don’t share those views?…

That support of social programs comes with a caveat, however: Taxes and the deficit ranked high on these voters’ list of concerns in 2012, including the economy and jobs. It is this mixed message that Republicans believe gives them an opening.

Health care was “popular in the past because you’ve had liberals selling it to a community that is underserved and under-insured,” said Korn. “Hispanics, just like the rest of Americans, are going to see their premiums go up and affect their small businesses.”