Mr. Graham argued that his party risks losing the presidency with its current hard line on immigration.

“We’ll lose,” he told USA Today. “I mean, we’ve got a big hole we’ve dug with Hispanics. We’ve gone from 44 percent of the Hispanic vote (in the 2004 presidential election) to 27 percent (in 2012). You’ll never convince me … it’s not because of the immigration debate.”

Mr. Graham also lashed out at Republicans who believe securing the border is the first priority.