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After a meeting last week, House Republicans emerged with a strategy that includes raising questions about Mr. Obama’s commitment to enforcing border security. The Senate’s immigration bill includes $40 billion toward additional security along the U.S.-Mexico border.

Given the persistent opposition among many House Republicans, Mr. Obama’s task of persuasion is “very difficult,” Mr. Barreto said.

“He needs to make [House Republicans] believe that they’re going to be in even worse standing and going to hurt the party’s reputation if they continue to stall or delay or block the bill,” he said. “I do believe that [House Speaker] John Boehner and [Majority leader] Eric Cantor and [Budget Committee Chairman] Paul Ryan and a couple of the more mainstream Republicans in the caucus are persuaded by the national reputation of the party and the future implications for the party nationally. The more that Obama is perceived as the hero and the more that the GOP is perceived as the villains on Spanish-language TV, the more that John Boehner’s going to be inspired to try to move the bill. They don’t want to concede any ground.”