Caving to pressure from Republicans in his caucus, House Speaker John Boehner has suggested he won't allow a vote on an immigration bill that lacks the support of a majority of Democrats and a minority of Republicans.

Boehner's requirement that a majority of House Republicans support a bill would seem to contradict his comments to ABC News' George Stephanopoulos in an interview last week.

"I've allowed the House to work with… well, more than any speaker in modern history, to the point where there are some bills that have passed with a majority of Democrats in favor, and a minority of Republicans," Boehner said back on June 11th. "And I've been criticized for it."

Asked whether he would do that with immigration reform, Boehner said was committed to an open process that allowed "all members" to "have a fair shot."

"What I'm committed to is a fair and open process on the floor of the House- so that all members- have an opportunity," he said. "It's not about what I want. It's about what the House wants. And my job is as speaker is to ensure that all members on both sides have a fair shot at their ideas."

But Boehner has sustained criticism from some members in his own caucus who believe that he should not violate the so-called "Hastert rule"-named after former Republican House Speaker Dennis Hastert-which requires that the Speaker only bring legislation to the floor if it has the support of the majority of his own party.

In a press conference Tuesday morning, Boehner appeared to relent to the criticism, saying plainly that on immigration, he would not defy his caucus.

"I also suggested to our members today that any immigration reform bill that is going to go into law ought to have a majority of both parties' support if we're really serious about making that happen," Boehner said. "And so I don't see any way of bringing an immigration bill to the floor that doesn't have a majority support of Republicans"

Requiring majority support from Republicans will make it much more difficult for a bill with a pathway to citizenship to pass through a House where many rank and file Republicans consider it de facto "amnesty."

Rep. Dana Rohrabacher (R-Calif.) warned Boehner in an interview with the conservative media site World Net Daily, that he could face rebellion from within the Republican ranks if he attempted to bring an immigration bill to the House floor without Republican support.

"I would consider that a betrayal of the Republican members of the House and a betrayal of the Republicans throughout the country," Rohrabacher said. "If Boehner moves forward … and permits this to come to a vote even though the majority of Republicans in the House - and that's if they do - oppose what's coming to a vote, he should be removed as Speaker."

(ABC News)