Republican lawmakers say they will not give President Barack Obama a "blank cheque" to combat an immigration crisis without additional policy changes.
House Speaker John Boehner and others want to speed up deportation of children migrating illegally.
Immigrant groups and some Democratic lawmakers have opposed such a change.
Since October 2013, more than 50,000 unaccompanied children have been detained trying to cross the southern US border, straining border resources.
The migrants - mostly from Central America - are believed to be driven north by a spike in violence in their home countries and incorrect rumours children will be allowed to stay if they make it across the border.
Republicans have blamed Mr Obama's immigration policy for the crisis, while the US president has harshly criticised House Republicans for not voting on an broad immigration bill passed by the Senate last year.
On Thursday Mr Boehner called the crisis "a problem of the president's own making".
"He's been president for five years! When is he going to take responsibility for something?"
The White House has previously said it was seeking additional authority to speed up deportations but did not include any policy change in a $3.7bn (£2.2bn) emergency spending request to combat the crisis.
The emergency funding includes money for the hiring of extra immigration judges, drone surveillance of the border, medical services and transportation costs, and expanding a border security task force in Central America.
More than 200 organisations advocating for immigrants signed a letter calling on Mr Obama to reconsider any policy changes they feared would limit the children's rights to asylum hearings.
Republicans say they want to amend a 2008 law intended to address human trafficking that guarantees deportation court hearings to children not from Canada and Mexico.
Democratic leadership in Congress have left open this could be part of a compromise on the spending request.
Where the $3.7bn will go
- $1.8bn to provide the appropriate care for unaccompanied children
- $1.1bn on detention and removal programmes
- $433m on border patrol and security
- $300m to Central American countries to repatriate and tackle root causes
- $64m on immigration courts, including hiring 40 additional judge teams
Source: White House
"It's not a deal-breaker," Democratic House Minority Leader Nancy Pelosi said.
"Let them have their face-saver. But let us have the resources to do what we have to do."
Such comments come a day after Mr Obama met Texas Governor Rick Perry and local leaders to discuss the crisis during a fundraising trip to the state.
The US president a said he had no "philosophical objection" to increasing the number of border patrol agents and repositioning them, a move Mr Perry has called for.
But Mr Obama said such action was held up by Congress's delay in approving his request for extra funding.