The measure would allow some troops to stay to train Iraqi soldiers and protect American diplomatic facilities.
But an influential group of liberal House Democrats, which wants a complete withdrawal of US troops from Iraq by December 31 this year, said they would oppose their leadership's plan and push for their own legislation.
Maxine Waters, a Democratic House representative from California and a member of the Out of Iraq caucus, said the bill allows Bush to decide if Iraq is meeting benchmarks.
Negotiations between Pelosi and the liberals could continue for the next two weeks under the planned timetable for passing the legislation.
Tough conditions
The House plan would place tough conditions on Bush implementing his plan to ramp up US forces fighting in Iraq.
He said in January he would send in 21,500 more combat forces, while Robert Gates, the US defence secretary, on Thursday approved a request by general David Petraeus, the US commander in Iraq, for an extra 2,200 military policemen in Baghdad.
Gordon England, the US deputy defence secretary, told legislators on Tuesday that between 4,000 and 7,000 more additional support troops were needed for the ongoing security operation in Baghdad.
Under the bill, Bush would have to give troops adequate rest from combat missions and ensure they had adequate training and equipment before sending them into war.
Bush could waive those provisions, but at some political risk.