Conservative lawmakers voiced objections, and Rep. John Fleming of Louisiana said Boehner told Republicans he was undecided about bringing the plan to the floor because he doesn't know if there are enough votes to pass it.
Rep. Mo Brooks, R-Ala, said he couldn't support it, complaining that Obama has turned the U.S. into the "world's sugar daddy."
Several GOP lawmakers said the House plan would cost about $1.5 billion, compared to Obama's original $3.7 billion request for more immigration judges, detention facilities and other resources to deal with unaccompanied kids.
A plan by Senate Appropriations Committee Chairwoman Barbara Mikulski would cost $2.7 billion.
But the biggest conflict between the House and Senate is not over costs, but policy instead.
Mikulski, D-Md., said she was omitting from her legislation any changes to a 2008 trafficking victims law that Republicans say has contributed to the crisis by allowing Central American youths to stay in this country indefinitely while awaiting far-off court dates.
Republicans are demanding changes in that law as the price for approving any money for the crisis.
"Modifications to the law can be done to expedite the process while ensuring proper protections are in place for the children who need them," said Rep. Kay Granger, R-Texas, who led Boehner's working group.
The result looks like a stalemate, with little time left to resolve it because Congress' annual August recess is just around the corner.
"Unfortunately, it looks like we're on a track to do absolutely nothing," Sen. John Cornyn, R-Texas, said.
It comes even as Homeland Security officials are pleading again for action, saying overstressed border and immigration agencies will run out of money in the next two months. "Doing nothing in Congress is not an option," Homeland Security Secretary Jeh Johnson said Tuesday.
More than 57,000 minors have arrived since October, mostly from El Salvador, Honduras and Guatemala.
The 2008 law guarantees them judicial hearings, which in practice allows them to stay in this country for years before any deportation can be carried out because of major backlogs in the immigration court system.
Republicans want the law changed so that unaccompanied Central American children can be treated like those from Mexico, who can be sent back by Border Patrol agents unless they can demonstrate a fear of return that necessitates further screening. Republicans say that's the only way to send a message to parents in the Central American nations that there's no point in sending their children on the arduous journey north.
White House officials have indicated support for such changes but have sent mixed signals, under pressure from immigration advocates who say they would amount to sending kids fleeing vicious gang violence back home to their deaths. Some Democrats initially were open to such changes but most are now strongly opposed.
"I'm very reluctant to change the law because I think these children face death, murder, vicious abuse, persecution, if they are returned," Sen. Richard Blumenthal, D-Conn., said.
Polls suggest the public is paying attention and demanding a solution, but lawmakers could not say where a compromise might lie.