Numbers from Latino Decisions (pdf) suggest that Republicans could gain Latino votes by moderating on immigration:

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Ezra explains the GOP’s thinking:

Two numbers explain why a rational Republican Party needs to do something dramatic on immigration: 27 percent and 2 percent. Twenty-seven percent is the percentage of the Latino vote Mitt Romney received in 2012, according to the exit polls. Two percent is the projected increase in the non-white electorate come 2016.

Chait summarizes the GOP plan as “go left on immigration, right on everything else”:

The key figures leading the way are Paul Ryan, the Republicans’ de facto leader, and Marco Rubio, perhaps its leading presidential candidate. The two have moved generally in tandem, with Rubio leading the way on immigration, but the whole party apparatus has jolted into action. Within days of the election, partisan barometers like Krauthammer and Sean Hannity had announced they had suddenly changed their mind and now favored comprehensive reform. Freed up to cut a deal, Rubio has thrown himself behind a reform plan liberals can happily accept, while he has steadily neutralized every source of conservative discontent. (Hardly a day goes by without some new Republican praising Rubio’s plan.) Crucially, Ryan himself has signaled support for Rubio. The party’s rapid embrace of immigration reform has been a sight to behold, a ruthlessly efficient exercise in partisan calculation.

Frum is skeptical that immigration reform will pay dividends:

The same senior Republican leaders who believed that Mitt Romney was winning the 2012 election now insist that immigration reform will deliver Hispanic votes. They’re just wrong about that.

Republicans face increasing difficulty with the Hispanic vote for pocketbook reasons. As the Hispanic electorate becomes less Cuban, more Mexican and Central American, it becomes less susceptible to GOP cultural themes. The claim that Hispanic voters are “natural Republicans” is based on nothing but wishful thinking, fortified by ignorance.

Bouie’s analysis is more nuanced:

The GOP’s best hope comes from assimilation. If Latinos follow the path blazed by Irish and Italians — and ethnic identity becomes less salient to political affiliation — then Republicans stand a chance at winning a sizable share of their votes. Already, you can see hints of this in Latino public opinion: First generation Hispanics are the most supportive of “big government” (81 percent to 12 percent), but third generation are the least supportive (58 percent to 36 percent). But even then, you have a situation where the bulk of younger Latinos are broadly supportive of government. In other words, no matter how you slice it, Latinos will remain a Democratic constituency. Republican outreach might work on the margins, but there’s no reason to expect anything more than that.