WASHINGTON — The fate of a Republican proposal to address a brewing immigration crisis along the U.S.-Mexico border was cast into doubt Wednesday after a tea-party senator lobbied against it to House members.

The effort by Sen. Ted Cruz of Texas, who made his pitch to a group of House Republicans in a closed-door evening meeting, marked another direct shot at attempts by Speaker John Boehner, R-Ohio, to deal with the influx of illegal immigrants arriving from Central America.

House members are preparing to vote Thursday on border legislation that would provide considerably less money than President Obama is seeking, while the Senate is moving forward with a larger Democratic proposal. With two competing border measures and little agreement between House and Senate leaders, figuring out how to pay for the unexpected surge is unlikely to be resolved before Congress’ summer recess begins Friday.

Even so, a defeat of border legislation in the House would deliver another embarrassing blow to Boehner and his leadership team, which has struggled to contain the party’s restive tea-party caucus. It also would serve as the latest example of Cruz wading into House affairs and working against the agenda of GOP leaders.

“The Obama White House should put Ted Cruz on the payroll,” said Rep. Peter King, R-N.Y., a vocal Cruz opponent. “We have a chance to pass a good bill, not a perfect bill. Boehner is working hard to get to 218 votes and yet there is Ted Cruz, telling us to do nothing. If he wants to come over and run for speaker, that’s fine, but otherwise he should stay over there in the Senate.”

Boehner has not promised victory on the immigration bill, suggesting Tuesday that Republicans had “a little more work to do.”

With most Democrats expected to vote against the measure, Boehner needs every GOP vote he can find and has been busy trying to win support from Reps. Steve King of Iowa, Louie Gohmert of Texas, Matt Salmon of Arizona and other conservatives.

At conference meeting Tuesday, Boehner announced that he would pare down his initial framework after hearing numerous complaints about its size and scope. On Thursday morning, he will meet again with GOP members to underscore the importance of passing his plan and giving the party a document that shows its ability to find consensus.

But Steve King, Gohmert and Salmon – along with Cruz and others – want House Republicans to defund Obama’s Deferred Action for Child Arrivals program, which has granted temporary relief for some children of illegal immigrants and is set for renewal this fall. Boehner has resisted the idea.

“The only way to stop the border crisis is to stop Obama’s amnesty,” Cruz said in a statement. “It is disappointing the border security legislation unveiled today does not include language to end Obama’s amnesty. Congress cannot hope to solve this problem without addressing the fundamental cause of it.”

Cruz, who is considering a 2016 presidential bid, also met with House Republicans over breakfast last week to discuss immigration and border issues.

An aide to Boehner declined to comment Wednesday on Cruz’s lobbying.