Two Republicans and two Democrats introduced a resolution that would require the president to announce by the end of this year a plan for withdrawing troops and steps for following through on that plan.
It is the first such resolution put forth by lawmakers from both parties, although an overwhelming number of Democrats and six House Republicans voted in 2002 against sending troops to Iraq.
In no uncertain terms, the administration regards it as a mistake to set a date for the withdrawal of U.S. troops, reports CBS News White House Correspondent Mark Knoller.
"It's important that we succeed and that means completing the mission," Press Secretary Scott McClellan said Thursday.
"We all want our troops home soon," he said, but not while they're still needed to thwart terrorists trying to derail Iraq's transition to democracy.
While many Democrats and some Republicans repeatedly have voted against continued funding for the war, there has been no concerted joint effort before to bring troops home.
A low-water mark of 41 percent of adults said in an Associated Press-Ipsos poll this month that they supported Mr. Bush's handling of the war in Iraq. And a Gallup poll released Monday found that six in 10 Americans say they think the United States should withdraw some or all of its troops from Iraq.