The measure, approved 221 to 205 by the Democratic Party-controlled House on May 10, would release nearly $43 billion in emergency war funding.
But the bill would require that Bush demonstrate in July that progress is being made on Iraqi security and self-government before any additional war funds are provided.
House speaker Nancy Pelosi called on Bush to reconsider his veto threat, saying the legislation was a fair bill that would enable Congress to hold the president responsible for his Iraq policy.
"This is a bill [Bush] should like -- it has his benchmarks, it asks for a progress report," Pelosi said. "He must have some confidence in what he is doing, and then leave it up to the Congress to make a judgement in July. What could be fairer than that?"
It remains unclear whether the Senate, where the Democrats hold a smaller majority, will be able to pass a similar bill.
Bush has threatened to veto any legislation that sets conditions on his ability to conduct the war. But in a sign of a possible compromise, the president has indicated he could consider a military funding bill that includes political and security goals for the Iraqi government to meet.
Before approving the war-funding bill, the House voted to reject a measure calling for the withdrawal of U.S. combat troops from Iraq within nine months. The measure, put forward by strongly anti-war Democrats, was rejected with a vote of 255 to 171.
(compiled from agency reports)