The House is expected to debate the removal of troops from Iraq all day on Thursday, before voting on the measure.
A redeployment would begin within 120 days and the president would be forced to report to Congress on why soldiers should stay in Iraq for "limited purposes".
A similar bill is also being debated in the Senate, but both approaches mirror earlier Democratic attempts to end the war which Bush vetoed.
Although Republican discontent is growing over Iraq, it is not clear if the Democrats have drawn enough former allies of Bush to clear the 60-vote hurdle in the 100-seat Senate which is needed for success.
Meanwhile, it is unclear what the much-anticipated Bush administration report on its so-called troop "surge" strategy will reveal.
ABC News, the US television channel, said the Bush administration would tell Congress that the Iraqi government merited "satisfactory" grades on eight of 18 benchmarks for political and military progress.
That conflicted with reports earlier in the week that Iraq would fail every single test in the report, which will set the table for a more definitive judgement on the "surge strategy" awaited in September.
"Hope and reality"
The political debate over the war took a new twist on Wednesday, as another Republican senator bowed to public opinion and declared she would back a Democratic bid to enforce troops withdrawals by next year.
Olympia Snowe, senator for Maine, said the US had arrived at a "crossroads of hope and reality" on the war and it was time to embrace "reality."
The White House, under withering political fire, has admitted unhappiness over the war had become the "central fact" of US politics, but rejected demands to reverse Bush's surge of 30,000 extra troops into Iraq.
Tony Snow, White House spokesman, said: "There's a lot of skepticism among Republicans ... they're getting an earful from constituents."
In a separate development, in a rare moment of unity on Iraq, the Senate voted unanimously to pass a measure censuring Iran for what it said was complicity in the killings of US soldiers.
Senator Joseph Lieberman, who framed the legislation, said: "Today's unanimous vote sends a strong, clear message from the entire Senate to the Iranians that we know what they are doing in Iraq, and they must stop."
Wednesday's amendment laid out what it said was evidence about proxy attacks by Iranian forces on US soldiers in Iraq and called for a regular US government report to Congress on Tehran's role in Iraq.