Sen. Ted Cruz (R-Tex.) has lobbied House members against a Republican proposal to address a brewing immigration crisis along the U.S.-Mexican border. (J. Scott Applewhite/AP)

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 (Tex.), 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 A. 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’s 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 T. 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.”

More than 57,000 unaccompanied minors have crossed into the United States from Mexico since October 2013. Here are 10 numbers to know about the growing humanitarian crisis. (Tom LeGro and Natalie Jennings/The Washington Post)

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 (Iowa), Louie Gohmert (Tex.), Matt Salmon (Ariz.) and other conservatives.

At a 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. But late Wednesday, GOP aides said that leaders were likely to allow a vote on a standalone bill that would defund DACA before voting to approve the border spending measure. If the bill to defund DACA were to pass, it wasn’t clear exactly how House leaders would merge the two proposals and send them to the Senate.

“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. Sen. Jeff Sessions (Ala.) has advised House Republicans to oppose Boehner’s plan as well.

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

The House Republican border measure would make it easier for the U.S. government to deport Central American minors who have entered the United States illegally and would provide $659 million in additional funding to federal agencies through the end of the fiscal year. The funding is significantly less than the $3.7 billion that Obama requested and less than the $1.5 billion initially floated by Boehner and his allies earlier this month.

“We’ve looked at it and looked at it and looked at it,” said Rep. Kay Granger (Tex.), who has led a GOP working group on the border issue. “It’s a good bill even if it’s not what everyone wants. It’s the right thing to do right now.”

But the White House threatened to veto the House measure Wednesday, saying the proposal “could make the situation worse, not better.” An administration statement said the legislation would “undercut due process for vulnerable children which could result in their removal to life threatening situations in foreign countries.”

The Democratic-backed Senate includes $2.7 billion in emergency funding for detaining, housing and eventually deporting the thousands of migrants who have crossed the Southern border in recent months.

Also Wednesday, House members easily passed a broad overhaul of the troubled Department of Veterans Affairs that would make it easier for veterans to seek medical care outside the government-run system. By a vote of 420 to 5, the House sent the bill to the Senate, where it is expected to be approved by the end of Thursday.

David Nakamura contributed to this report.