So now the bargaining begins.
Both sides have powerful incentives to reach an agreement. Democrats are eager to show voters they can use their congressional majority to deliver concrete relief. Bush wants to protect his reputation - and Republican candidates' chances in November - by responding to a leading concern.
The same impulses were at work earlier this year when House Speaker Nancy Pelosi, Democrat of California, teamed with Treasury Secretary Henry Paulson and Representative John A. Boehner, Ohio Republican and the minority leader, to reach an uncommonly smooth and swift agreement on a $168 billion economic aid package with tax rebates for most American wage-earners.
The White House has been hostile to the homeowner aid plan by Representative Barney Frank, which would offer government insurance on $300 billion in new mortgages so that strapped homeowners could refinance into more affordable, fixed-rate loans.
Frank, a Newton Democrat, argues that his plan, which passed the House on Thursday, will prevent hundreds of thousands of foreclosures and help stabilize the housing market. Bush calls it a burdensome bailout for lenders that would open taxpayers to too much risk.
Behind the rhetoric, however, the administration and Frank agree on the central concept of a homeowner rescue. Both want the Federal Housing Administration, the Depression-era mortgage insurer, to help more borrowers refinance into loans they can afford. Bush has twice relaxed FHA standards to allow more "workouts" for people facing default because of economic hardships.
When it came time to vote on Frank's bill, 39 Republicans broke with Bush to back the plan, many of them saying they were shelving their philosophical objections to a big-government intervention because their constituents were hurting.
"In my view, this is the only train leaving the station," said Representative Ric Keller, a Florida Republican.
Lobbyists and congressional aides working on the measure see potential for a compromise that might shrink the size and scope of the package - now estimated capable of helping 500,000 homeowners, at a cost of $2.7 billion over the next five years - but keep its essential elements intact. Frank, who has a strong working relationship with Paulson and Federal Reserve chairman Ben Bernanke, is uniquely positioned to reach such a deal.
The measure still must make its way through the closely divided Senate, where virtually everything requires a bipartisan majority of 60 votes to pass.