Sen. John Warner and six other Republicans told leaders from both parties in a letter Wednesday that "the current stalemate is unacceptable to us and to the people of this country."
The senators warned they would attach the resolution against the troop increase to every piece of legislation they can in an effort to force a debate.
"The war in Iraq is the most pressing issue of our time. It urgently deserves the attention of the full Senate and a full debate on the Senate floor without delay," the letter said.
Warner announced the joint letter on the floor of the Senate. Some of those who signed it earlier had voted to bar debate.
Five other Republicans who oppose the troop increase and voted to bottle up the measure signed the letter: Sens. Warner, Chuck Hagel, Olympia Snowe, Gordon Smith and George Voinovich.
Jim Manley, spokesman for Sen. Harry Reid, leader of the Senate's Democratic majority, said he hoped the letter signified that the Republicans "have had a change of heart and will be willing to vote for their own resolution in the future."
The bipartisan resolution expressed dissatisfaction with Mr. Bush's plan to send 21,500 more troops to Iraq and identified benchmarks that the Iraqi government should meet. It fell 11 votes short of the 60 required to move the debate forward.
In a bid to attract more GOP support, Warner had added a section promising to protect funding for troops in combat — a promise many House Democrats do not want to make.
House Democratic leaders hope to have a vote on a resolution of their own by next week.
"What we put forth will have clarity and consensus and it will be something we hope that the president will hear and it will have the support of Democrats and Republicans," said House Speaker Nancy Pelosi, D-Calif.
Democrats are said to be drafting a short and simple measure to make it easier for GOP lawmakers to break ranks, reports CBS News correspondent Susan Roberts.
House Democratic leaders met privately Wednesday to discuss the proposal, which they planned to present to members during a Thursday caucus meeting.
Republican leaders, who are seeking their own alternative measure that would set benchmarks for the Iraqi government, said they anticipate some GOP defections to the nonbinding Democratic resolution.
"I don't think it'll be a pure party-line vote," said Rep. Adam Putnam, R-Fla., chairman of the House Republican Conference
Also Wednesday, Defense Secretary Robert Gates and Marine Corps Gen. Peter Pace, chairman of the Joint Chiefs of Staff, told lawmakers they did not think debate in Congress would hurt the morale of troops in combat.
Less than two weeks ago, Gates said a congressional resolution opposing the Iraq strategy undercuts U.S. commanders in Iraq and "emboldens the enemy."
"As long as this Congress continues to do what it has done, which is to provide the resources for the mission, the dialogue will be the dialogue, and the troops will feel supported," Pace told the House Armed Services Committee.
Gates said troops understand members of Congress want to find the best way to win the war. "I think they're sophisticated enough to understand that that's what the debate's really about," he said.