Speaker John Boehner John Andrew BoehnerLongtime House parliamentarian to step down Five things we learned from this year's primaries Bad blood between Pelosi, Meadows complicates coronavirus talks MORE (R-Ohio) on Tuesday told GOP lawmakers that he would adhere to the “Hastert Rule” on immigration reform and not hold a vote without the support of a majority of the caucus.

"I don't see any way of bringing an immigration bill to the floor that doesn't have a majority support of Republicans," Boehner John Andrew BoehnerLongtime House parliamentarian to step down Five things we learned from this year's primaries Bad blood between Pelosi, Meadows complicates coronavirus talks MORE told reporters following a closed-door House GOP conference meeting, where he offered a similar message to his members.

However, Boehner may have given himself an out if the House and Senate both pass immigration measures and then work on a conference agreement reconciling their approaches.

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Asked by a reporter if he would bring up a conference report on immigration with less than a majority support from House Republicans, Boehner responded with a quick, "We'll see when we get there."

In terms of a conference report, a source familiar with Boehner's thinking explained to The Hill that it was unclear if both chambers will move comprehensive immigration reform bills that could be considered by a conference committee, let alone whether House and Senate conferees could agree on a compromise of such proposals.

Boehner is skeptical that Senate Democrats will agree to tough border security measures needed for GOP support on any comprehensive bill, sources in the meeting said.

In discussing the issue with his conference, Boehner argued he would only bring up a bill supported by less than a majority of his conference if he has no other leverage. And he said he did not envision such a "scenario" when it comes to immigration.

"The only time any Speaker allows a major bill to pass without a majority of the majority is when there is zero leverage," Boehner told his rank-and-file members, according to a source in the room.

Boehner noted that longtime GOP Speaker Dennis Hastert (Ill.) had to do this with campaign finance reform, and that former Speaker Nancy Pelosi (D-Calif.) had to do it with Iraq war funding.

Boehner conceded to doing it a "couple times, such as the fiscal cliff and hurricane relief, where we had no leverage and we faced a worse alternative — politically or in terms of policy," according to the source who attended the meeting.

He then said: "Let me be clear: immigration is not one of these scenarios," the source said Boehner told his colleagues. "We have plenty of leverage. And I have no intention of putting a bill on the floor that will violate the principles of our majority and divide our conference."

"One of our principles is border security. I have no intention of putting a bill on the floor that the people in this room do not believe secures our borders. It’s not gonna happen."

Conservatives have been pressuring Boehner to adhere to the informal GOP rule named for Hastert that requires support of the “majority of the majority” for consideration of bills by the full House.

Boehner's decision is likely to complicate efforts to get an immigration bill to President Obama's desk.

A comprehensive bill in the Senate appears to be on its way to passage as early as this month, though Senate Republicans are battling to include tougher anti-illegal immigration security measures in it.

The Senate bill is opposed by many conservatives in the House, however, who believe it provides "amnesty" to illegal immigrants without doing enough to stop illegal immigration.

Immigration reform advocates believe a strong bipartisan vote in the Senate could increase pressure on the House, but it is clear Boehner will be under pressure from his own conference not to bend to the Senate.

On Monday, Rep. Dana Rohrabacher (R-Calif.) warned that Boehner should lose his gavel if he moved forward on immigration without majority support, saying it would be a “betrayal” of the party.

“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,” he said in an interview.

Asked by reporters if he agreed with Rohrabacher’s assessment, the Speaker considered the question and replied, “Maybe.”

Boehner acknowledged the concerns of conservatives about the Senate immigration bill being debated this week, warning Obama and Senate Democrats to "get serious" about border security and calling the Gang of Eight bill “weak” on that issue.

Boehner said he told GOP members that any “reform bill that is going to go into law ought to have a majority of both parties' support — if we are really serious about making that happen."

"I think this immigration issue has been kicked around this town for 15 years — that's why I said the day after the election that it's time for Congress to do its work,” Boehner said.

“I'm increasingly concerned that the White House and Senate Democrats would rather have this as an issue in the 2014 election rather than a result. ... On the House side, I made clear to our members that we're going to have a discussion about this on July the 10th — we're going to lay all this out and listen to what the members have to say," he added.

The Congressional Hispanic Caucus will also meet privately with Boehner on Wednesday.

A leading Democratic advocate for immigration reform, Rep. Luis Gutiérrez (Ill.), said the group would not necessarily pressure the Speaker to force a vote if he lacks support from a majority of Republicans.

"I don't know that he can do it," Gutiérrez said on Tuesday.

"Here’s what I believe he can do, and I hope he will embrace: a bipartisan route on comprehensive immigration reform," he said.

Gutiérrez, a member of the bipartisan House immigration group, nonetheless said he believed a majority of the House would back a comprehensive bill.

"I hope in the end that this doesn’t become a partisan issue and that they allow a majority of the House of Representatives, who really represent a majority of the people of the United States of America, to vote their will," he said.

"I will bet every ounce of my reputation, my political reputation for being able to count votes, that the votes exist in the House of Representatives. The votes exist. There are more than [218] for comprehensive immigration reform."

A GOP special conference is scheduled to take place after the House Judiciary and other committees mark up various immigration-related measures.

House Judiciary Committee Chairman Bob Goodlatte Robert (Bob) William GoodlatteNo documents? Hoping for legalization? Be wary of Joe Biden Press: Trump's final presidential pardon: himself USCIS chief Cuccinelli blames Paul Ryan for immigration inaction MORE (R-Va.) has said he will not follow the Senate’s comprehensive approach, instead moving immigration legislation piecemeal.

—Russell Berman contributed

This story was posted at 11:05 a.m. and last updated at 2:17 p.m.