If Bush could not certify progress towards ending violence in Iraq, withdrawals would begin in July of this year and be complete by December 31.
A House aide said that Bush would not be able to waive or extend that deadline if the proposal becomes law.
But senate opposition could stop the legislation.
Republicans in the US congress and the Bush administration denounced the Democratic plan to withdraw US troops from Iraq next year.
"Peace in Iraq will happen when citizens realise that they can express their ideas and views more effectively... than with violence"
JBernar5, Toledo, USA
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Tony Snow, a White House spokesman, said: "It is apparent, once you look at the details of this proposal that the chief aim of Democratic leaders was to get Democrats happy, rather than the more important goal of providing the funding and flexibility generals need to succeed in their mission in Iraq."
Dan Bartlett, a presidential counsellor, called the plan "a non-starter for the president." He said it would "handcuff" US generals.
Democrats at odds
Pelosi, a California Democrat, wants the proposal approved next week in the House Appropriations Committee as part of a $100bn bill to continue funding the US wars in Iraq and Afghanistan.
If so, the full House would debate the measure the following week.
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.
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.