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But he also said the support likely would top out at about 35 percent, which would be up from the 27 percent support for Mr. Romney this year, and the 31 percent support for Sen. John McCain’s 2008 presidential run. But it would be less than the 40 percent or so that President George W. Bush won in his 2004 re-election bid.

Alfonso Aguilar, a Bush administration official who is now the executive director of the Latino Partnership for Conservative Principles, said the Republican Party has no choice but to try to do better.

“You either engage them, or you become an irrelevant party and movement,” he said. “We already cannot win California and New York. If we continue like this, it’s [next] going to be Florida, Texas, Arizona. That’s it — Republicans will never win another national election.”

He said even though Hispanics use social services more than native-born whites, many of them don’t want to be on welfare, and would vote their aspirations rather than their circumstances if the Republicans didn’t turn them off.

“These are people who are coming here because of our system of liberties and opportunities. They come here, they’re very entrepreneurial, they’re opening businesses three times as fast as the country as a whole,” he said. “When you ask them should government provide more services, they’re probably going to say ‘yes.’ That’s a basic answer. But I think it reflects a view of their home country, where government is huge. But at the same time, I don’t think they want to see a government the size of their home country, where the economy was unsuccessful.”

He said some of the communities that have the deepest, long-term ties to the U.S. — such as Mexicans in the Southwest or Puerto Ricans, who saw a major wave of immigration decades ago to places such as New York and Chicago — do match Democrats’ message better because they “sadly fell through the trap of entitlements.”

But he said for newer immigrants in places such as North Carolina or Georgia, “I don’t think they have that mentality.”

Mr. Camarota’s research data does, indeed, suggest that U.S.-born Hispanics who are third generation or more are worse off on just about every measure than second-generation Hispanics.

Both Mr. Aguilar and Mr. Segura said it will take more than an appeal on immigration to open up Republican channels to Hispanic voters.

But Mr. Aguilar said tackling immigration from a conservative standpoint of family reunification and economic opportunities will at least give the party a chance to be part of the conversation.

“We’re not looking right now to win the majority of Latinos. We can get to that 10, 15 years from now. But right now it’s about getting, consistently, 40 percent,” he said.