WASHINGTON — All but drowned out by Republicans’ clamorous opposition to President Obama’s executive action on immigration are some leaders who worry that their party could alienate the fastest-growing group of voters, for 2016 and beyond, if its hottest heads become its face.

They cite the Republican Party’s official analysis of what went wrong in 2012, the presidential-election year in which nominee Mitt Romney urged Latinos here illegally to “self-deport.”

“If Hispanics think that we do not want them here,” the report said, “they will close their ears to our policies.”

“Both the president and the Republican Party confront risks here,” said Bill McInturff, a Republican pollster. While the danger for Mr. Obama is “being perceived as overstepping his boundaries,” Mr. McInturff said, “the Republicans’ risk is opposing his action without an appropriate tenor, and thereby alienating the Latino community.”