The Clinton-Byrd proposal, which is not expected to pass in the Senate, forces the administration to seek new authorization for the war, declaring a "new mission" in Iraq that Congress would have to approve.
Another bill enjoying bipartisan support, co-sponsored by Colorado Democrat Ken Salazar and Tennessee Republican Lamar Alexander, says the United States should enact the policy recommendations of the Iraq Study Group as part of its future Iraq policy.
The Iraq Study Group, a bipartisan commission appointed by President Bush and led by Republican James Baker and Democrat Lee Hamilton, recommended the United States should turn the combat mission over to the Iraqis and work toward a withdrawal of U.S. forces from the region during spring 2008, while allowing to maintain long-term presence there.
Salazar's bill would make the 79 recommendations of the Baker-Hamilton Commission a U.S. policy.
Ahead of the 23-page surge status report expected from the White House later this week, National Security Adviser Stephen Hadley met Wednesday with GOP lawmakers for the second straight day.
Hadley's visit to Congress came on the heels of several Republican defections on Iraq last week, like key GOP Sen. Pete Domenici, R-N.M., a backer of Salazar's amendment to enact the recommendations of the Iraq Study Group.
Said Domenici last week, "I do not support an immediate withdrawal from Iraq or a reduction in funding for our troops, but I do support a new strategy that will move our troops out of combat operations and on the path to coming home."
Republicans at Wednesday' meeting offered Hadley lukewarm support for the administration's policy and expressed concern over the lack of progress in Iraq. Dissenting GOP voices told Hadley privately they didn't want the president to wait until September — when Congress reconvenes — to change course in Iraq.
In Cleveland this week, the president put his faith in his general who isn't due stateside till September.
"I call upon the U.S. Congress to give Gen. David Petraeus a chance to come back and tell us whether his strategy is working. And then we can work together on a way forward," Bush said.
Members said Wednesday that Hadley did not indicate whether or not the president was considering switching gears.
The Associated Press contributed to this report.