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“Obama voters chose dependency over Liberty. Now establishment R’s want citizenship for illegals. You can’t beat Santa Claus with amnesty,” Mr. King tweeted Thursday evening.

But other prominent conservatives have said the election should prompt rethinking of the GOP’s stance on immigration. Radio show host Sean Hannity said Thursday that he now supports a pathway to citizenship, and columnist Charles Krauthammer said Republicans should unapologetically embrace amnesty.

“Yes — amnesty. Use the word,” Mr. Krauthammer wrote Thursday in his election analysis.

Friday morning, at his weekly press conference, Mr. Boehner clarified his intentions on the issue.

“I’m not talking about a 3,000-page bill,” he said. “What I’m talking about is a common sense, step-by-step approach that secures our borders, allows us to enforce our laws and fix our broken immigration system.”