The 221-205 vote, largely along party lines, sent the measure to a cool reception in the Senate, where Majority Leader Harry Reid, D-Nev., is seeking compromise with the White House and Republicans on a funding bill.
Under increasing political pressure from Republicans, Mr. Bush also signalled flexibility, offering to accept a spending bill that sets out standards for the Iraqi government to meet.
"Time's running out, because the longer we wait the more strain we're going to put on the military," said the president, who previously had insisted on what he termed a "clean" war funding bill.
Mr. Bush and key lawmakers have stepped up expressions of frustration with the government in Baghdad in recent weeks, and Iraqi Deputy Prime Minister Barham Saleh spent his day in a series of meetings with key senators appealing for patience.
In a brief interview with The Associated Press, Saleh said the purpose of the meetings was to convey the "imperative of success against terrorism and extremism" in the Middle East.
Despite Mr. Bush's ability to sustain his vetoes in Congress — the House upheld his rejection of a troop withdrawal timetable last week — Democrats called for votes on two separate bills Thursday that challenged him on the war.
"Democrats are not going to give the president a blank check for a war without end," vowed House Speaker Nancy Pelosi of California.
The key is deciding what happens if Iraqis don't meet the standards set, reports CBS News chief White House correspondent Jim Axelrod.
Democrats want consequences for failure, but the president won't go that far yet, although he has instructed top aides to negotiate with Congress on that point, Axelrod reports.
The first measure would have required the withdrawal of U.S. combat forces from Iraq within nine months. It fell, 255-171, with 59 Democrats joining almost all Republicans in opposition.
"This war is a terrible tragedy and it is time to bring it to an end," said Rep. James McGovern, leading advocate of the bill to establish a nine-month withdrawal timetable. "For four long, deadly years, this administration and their allies in Congress have been flat wrong about Iraq," said the Massachusetts Democrat.
Republicans argued that a withdrawal would be disastrous.
"Now is not the time to signal retreat and surrender. How could this Congress walk away from our men and women in uniform," said Rep. Jerry Lewis, R-Calif.
A few hours later, the House passed legislation providing funds for the war grudgingly, in two installments. The first portion would cover costs until Aug. 1 — $42.8 billion to buy equipment and train Iraqi and Afghan security forces.
Under the bill, it would take a summertime vote by Congress to free an additional $52.8 billion, the money needed to cover costs through the Sept. 30 end of the fiscal year.
"We reject that idea. It won't work," the president declared after a meeting with military leaders at the Pentagon.
Democratic officials, speaking privately, said Pelosi had agreed to allow the vote on the withdrawal measure in the hope that her rank-and-file would then unite behind the funding bill.
But in an increasingly complex political environment, even that measure was deemed to be dead on arrival in the Senate, where Democrats hold a narrow advantage and the rules give Republicans leverage to block legislation.