Speaker John A. Boehner told his restive conference Tuesday that he doesn’t see “any way” that an immigration bill would come to the House floor without their backing — amid an internal revolt over Boehner’s refusal last week to rule out bringing a bill to the floor that would be passed mostly by Democrats.

“I don’t see any way of bringing an immigration bill to the floor that doesn’t have a majority support of Republicans,” the Ohio Republican said in a press conference Tuesday morning.

Boehner said he told his conference that the bill ought to have majority support among both Democrats and Republicans.

Asked about Rep. Dana Rohrabacher’s remark that if he broke the Hastert rule on immigration he could lose his speakership, Boehner paused.

“Maybe,” he quipped, to laughter in the room.

Boehner did not answer a question on whether he would require a majority of the majority on a final conference report on an immigration bill. “We’ll see when we get there,” he said.

His remarks seemed just shy of ruling out breaking the “Hastert rule,” or the general practice that bills on the floor have support from a majority of the majority. An insurgent group of Republicans last week began circulating petitions in hopes of codifying the rule.

He said he was increasingly concerned that Democrats want immigration as an issue in the 2014 elections rather than a law. He also said border security provisions would have to be strengthened considerably from what the Senate is considering.

“We know that border security is absolutely essential if we’re going to give people the confidence that we can do the rest of what needs to be done,” he said. “And I frankly think that the Senate bill is weak on border security, I think the internal enforcement mechanisms are weak and the triggers are almost laughable. So if we’re serious about getting an immigration bill finished I think the president and Democrats ought to reach out to Republican colleagues to build broad bipartisan support for the bill.”

Emma Dumain contributed to this report.