But the extra funding comes with a clause that the White House sees as unacceptable: a requirement that US troops start withdrawing from Iraq in October.
If, as expected, the bill passes both houses of congress, it would set the stage for the first major showdown with the Bush administration since the Democrats seized control of Capitol Hill after last November's mid-term elections.
"For the first time, the president will have to face up, will have to be accountable for this war in Iraq. And he does not want to face that reality," Nancy Pelosi, the speaker of the House of Representatives, said on Tuesday about the bill.
"Tonight, the House of Representatives voted for failure in Iraq and the president will veto its bill," Dana Perino said, White House spokeswoman, said.
The extra funds in the bill, above that sought by the White House, is meant to respond to needs of veterans returning from war who at times have faced poor quality medical care.
Ahead of the promised presidential veto, expected next week, Republicans and Democrats have been scrambling to squeeze the maximum political mileage out of the debate, stepping up their rhetorical jabs.
"Our troops are mired in a civil war with no clear enemy and no clear strategy for success,'' said Steny Hoyer, majority leader in the House of Representatives.
Backed by surveys showing the Iraq war is hugely unpopular, which helped their party forge their majority in congress, Democrats argue there is a general desire for a new direction in Iraq strategy.
"The new direction, Mr President, did not mean escalation of the war. It meant bringing stability to the region, and bringing our troops home safely and as soon as reasonably possible," Pelosi said on Tuesday.
Unlike an earlier version the house passed last month, this bill would not set a firm date for all US combat troops to leave the war. Instead, a nonbinding March 31 date for finishing the withdrawal merely would be a "target".
The bill would allow some US troops to stay in Iraq beyond March to continue training Iraqi soldiers, protect US facilities and to conduct targeted counter-terrorism missions.
The White House has slammed as political expediency the Democrats' stated desire to implement what the US public wants in Iraq.
It has branded senator Harry Reid, the majority leader in the senate, a defeatist for saying that the war was already lost at least on the military front.
In rare remarks delivered at the US Capitol after meetings with fellow Republicans, Dick Cheney, the vice-president, criticised the compromise bill hammered out late on Monday between house and senate Democratic leaders.
"What's most troubling ... is his defeatism," Cheney said of Reid. "The timetable legislation that he is now pursuing would guarantee defeat."
Reid hit back that Cheney was the administration's "attack dog" with single-digit public support levels.
Ahead of Wednesday's vote Bush dispatched his senior general in Iraq, David Petraeus, and other top officials to congress to make the case that his administration's strategy needs more time to work.
Afterward, Petraeus told reporters that sectarian murders in Baghdad were only a third of what they had been in January, before Bush began sending in additional US forces.
He added that progress in the troubled western Anbar province was "breathtaking", and he thought Nuri al-Maliki, the Iraqi prime minister, was "doing his best" at leading the country.
Still, "the ability of al-Qaeda to conduct horrific, sensational attacks obviously has represented a setback and is an area in which we're focusing considerable attention," Petraeus said.