The proposed legislation provides $659 million, trimmed from $1.5 billion and $1 billion proposals floated last week to appease conservatives reluctant of providing any additional funding.
Obama originally requested $3.7 billion to bolster security at the border, hire more immigration judges to more quickly work through the backlog of cases and take care of children already in the U.S.
Democrats in the Senate proposed a $2.7 billion plan that also included funds to help combat wildfires in the drought-stricken West and additional aid to Israel to support the Iron Dome defense system that has been successful in blocking rockets fired from Gaza.
The House plan provides for neither of those additional issues, and offers $405 million towards increased law enforcement and processing, while providing just $197 million to provide shelter and care for the children in government custody, less than $0.11 on the dollar from the administration's request.
Also in the bill will be a requirement that the funds must be spent by September 30, the end of the fiscal year.
"I think there is sufficient support in the House," House Speaker John Boehner, R-Ohio, told reporters after the weekly meeting of the House Republican Conference. "We have a little more work to do."
And while Democrats seem posed to stomach the shortened window and shrunken funding, they have balked over a tweak to a 2008 anti-trafficking law that would allow children from noncontiguous countries, like those from Mexico and Canada, to be turned around immediately and sent home.
House Minority Whip Steny Hoyer, D-Md., slammed the "my way or the highway"-style legislating by Republicans that would see the House pass a legislative change without first holding hearings and debate and then leave the Senate with little time to respond to a bill and for the two sides to reach a compromise ahead of the August recess.
"I don't think it would be a poison pill, the $659 [billion] as opposed to the $1.5 or $2.7 [billion]... it's bad policy but it wouldn't, in it of itself, kill it for the Senate," Hoyer said. "But clearly, if you're going to make policy changes and do it any kind of considered way... there's nobody who believes in my view that the Senate would be able to consider this in the eight hours or 16 hours that they have left this week."
Instead, Hoyer suggested, Congress should pass the funding "to meet the challenge that everybody admits exists," then deal separately with whether to make legislative changes. The White House had signaled its willingness to change the law, then backtracked and said it believed it could speed up processing of children under an emergency provision in the existing law.
Some 52,000 unaccompanied children, mostly from Guatemala, El Salvador and Honduras have entered the U.S. this year alone. That figure has doubled since last year, a jump many Republicans have blamed on misinformation over the 2012 Deferred Action for Childhood Arrivals program, which grants some undocumented immigrants relief from deportation.
While House Republicans conceivably could, as it often does, pass its version of the bill without any Democratic votes, some conservatives have made the repeal of DACA a condition of their support and could vote against Tuesday's proposal.