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Exactly what that pathway will look like and how many Republicans will support it remain to be seen. But in light of November’s election results, Republican leaders acknowledge that they must support some policies — a pathway to citizenship or perhaps something else — that ingratiate the party with Hispanics.

“How can we be a party of growth, of free enterprise, of prosperity, but not be the party of immigration?” said Carlos Gutierrez, a campaign adviser for Mr. Romney who has since founded the super PAC Republicans for Immigration Reform.

“What [Hispanics] sense is that we don’t welcome them. We have to be the party that celebrates immigration,” he said on “State of the Union.”

Mr. Romney’s vice presidential pick, Rep. Paul Ryan of Wisconsin, also said immigration reform can be accomplished this year. He said Republicans must embrace “a balance between respecting the rule of law and adhering to the reality of the day.”

But, like others in his party, he fears that Democrats may try to widen their lead among Hispanic voters rather than find common ground with Republicans.

“The president has a big speech coming up. The question many of us are asking, Republicans and Democrats: Is he looking to play politics or does he want to solve the problem? We don’t know the answer to that,” Mr. Ryan said on NBC’s “Meet the Press.”