House Republican conservatives warned Wednesday they would move to oust Speaker John Boehner if he puts an immigration reform bill up for a vote that doesn’t have the support of a majority of GOP members.

“There gets to be a time when there is a straw that breaks the proverbial camel’s back,” Rep. Matt Salmon, R-Ariz., said at a monthly question and answer event featuring a group of conservative lawmakers.

Salmon, along with Reps. Raul Labrador, R-Idaho; Tim Huelskamp, R-Kansas; and Rep. Tom McClintock, of California, said they do not support the Gang of Eight comprehensive immigration reform plan now advancing in the Senate, despite the passage of an amendment that would at 20,000 border agents and other border security measures to the bill.

Boehner, R-Ohio, told Republicans today he will not take up the Senate bill, repeating a pledge he made in writing last month, and instead plans to move House-authored legislation. But many in the conservative wing of the GOP conference fear Boehner will allow a bill on the floor that includes legalization or a pathway to citizenship for the nation’s 11 million illegal immigrants but does not include the level of border security they believe is needed to stop illegal migration.

Such a bill could pass with mostly Democratic support and a fraction of the Republican conference backing it.

Salmon said if Boehner chooses that path, “We would probably be frustrated to the point of looking for new leaders.”

McClintock said that if Boehner allows an immigration reform bill on the House floor that requires mostly Democrats to pass it, “it would be cause for removal, in my judgement.”

“The American people put us in the majority in the House and I think we need to act like a majority,” Salmon added later.

Boehner has been criticized recently by GOP lawmakers for bringing bills to the floor that lack adequate GOP support and are largely backed by Democrats.

On the opening day of Congress in January, a small group of conservative lawmakers tried to block Boehner’s election to the speakership. The plan fizzled, but it underscored the growing sense of discontent with his leadership among the party’s right flank.

Earlier this month, Boehner said he did not foresee bringing a bill to the floor unless a majority of House Republicans support it. The House is now moving a series of smaller bills that deal individually with expandedg visa programs, increased border security and an employment verification system for employers.

None of the conservative lawmakers at Wednesday’s event said they back the new Senate amendment, authored by Sens. Bob Corker, R-Tenn., and John Hoeven, R-N.D., to boost border security, including adding 20,000 more border agents. The lawmakers said they object to it because it does not require full implementation before clearing the way for illegal immigrants to obtain citizenship.

“If you flipped it, where border enforcement has to be in place before legalization,” Labrador said. “I would be more supportive.”

The House, added Huelskamp, “will have a different approach. We are going to do border security first. If the Senate bill passes, I think the phone lines on the House side are really going to light up.”

Corker disputed the claim by conservatives about his amendment.

“If the Hoeven-Corker amendment becomes law, 10 years must pass and there must be 700 miles of pedestrian fencing and 20,000 additional border patrol agents along the southern border before a Green Card is issued to those with RPI status,” Corker said.