Bush has said he will not be swayed by Congress or public criticism.
'Not giving up'
The White House was busy trying to rally support for Bush's plan.
Some senators who visited the White House and met Stephen Hadley, the national security adviser, on Wednesday said they may come up with their own counter-resolution supporting the president's plan.
John Cornyn, a Republican senator, said: "There were senators who were suggesting that approach... of not giving up in Iraq."
The resolution was drafted by Biden, who is also the chairman of the Senate Foreign Relations Committee, Democrat Carl Levin, the Armed Services Committee chairman, and Republican Chuck Hagel, a long-time war critic.
Hagel told a press conference: "I will do everything I can to stop the president's policy as he outlined it Wednesday night. I think it is dangerously irresponsible."
But Tony Snow, a White House spokesman, said the non-binding resolution would not affect Bush's plan.
He said: "The president has obligations as a commander in chief. And he will go ahead and execute them."
"I made my decision and we're going forward," Bush told CBS "60 Minutes" in an interview broadcast on Sunday.
"I will do everything I can to stop the president's policy as he outlined it Wednesday night. I think it is dangerously irresponsible"
Chuck Hagel, Republican senator
Levin said the resolution says "we do not support increased troops, deeper military involvement" and calls for shifting the mission of US troops from combat to training, counterterrorism and protecting Iraq's territorial integrity.
He said it also calls for "the greater engagement of other countries in the region in the stabilisation and reconstruction of Iraq".
The resolution does not call for a withdrawal of troops or threaten funding of military operations, as many Democrats have suggested.
Instead, the legislation says the US should transfer responsibility to the Iraqis "under an appropriately expedited timeline".
Hagel denied that the sponsors were defeatist, were trying to assign blame to Bush, or had political motives, even though he and Biden are expected to run for president.
He said: "No one wants to see this country humiliated, defeated, or in any way lose its purpose.
"There is no moral high ground that one group of senators has over the others."
Biden said he hoped to have a hearing on the resolution opposing Bush's plan in the Foreign Relations Committee next week. It was unclear how soon after that a Senate floor vote could take place, but it was not expected to happen before Bush delivers his annual State of the Union address to Congress on Tuesday.
"The only way to get the president to change course is to show him that members of his own party oppose him"
US Senate aide
Steny Hoyer, the Democratic leader of the House of Representatives, said he also expects the House to pass a resolution by a significant bipartisan majority expressing disapproval of the troop increase, but was waiting for the Senate to go first.
With polls showing that most Americans disapprove of the troop increase, the resolutions would force Republicans to reveal publicly where they stand on Bush's strategy and could further isolate the White House.
One Senate aide said: "The key here is bipartisanship.
"The only way to get the president to change course is to show him that members of his own party oppose him."
Bush will try to persuade Republicans to stick with his strategy. The White House says a resolution could send a signal to the world that the US is divided on the war.
A poll released this week by the Pew Research Centre said 61 per cent of US citizens opposed Bush's plan, while 31 per cent were in favour of it.
Senator Hillary Rodham Clinton, a potential presidential candidate, said the US should cut funds for protecting Iraq's government leaders and equipping its military if the country's Shia leaders fail to give Sunnis a greater role in government.
The New York Democrat also said she opposed sending more troops to Iraq, in part because it would siphon off US military strength in Afghanistan.