The nonbinding resolution would put the Senate on record as saying the U.S. commitment in Iraq "can only be sustained" with popular support among the American public and in Congress.
"I will do everything I can to stop the president's policy as he outlined it Wednesday night," said Sen. Chuck Hagel, a Nebraska Republican and potential 2008 presidential candidate who joined Democrats at a news conference on the resolution.
Later Wednesday, a second Senate Republican, Olympia Snowe, a moderate from Maine, joined Democrats opposing the president.
Mr. Bush is concerned enough about losing even more Republican support that he met privately during the afternoon with selected GOP members of Congress, presumably to nudge back them in his direction, reports CBS News correspondent Sharyl Attkisson.
"It is not in the national interest of the United States to deepen its military involvement in Iraq, particularly by escalating U.S. troop presence in Iraq," the resolution says.
Hagel, who helped draft the proposal with Democratic Sens. Joe Biden and Carl Levin, called the resolution a "genuine bipartisan effort."
However, some Republicans denounced it as a political ploy to embarrass Bush. Sen. John Cornyn, a Bush supporter, predicted the resolution would fail.
"If my Democrat colleagues are truly opposed to the mission in Iraq, then as the new majority in Congress they should schedule a serious debate and a vote on cutting off funding for our troops," said Cornyn, a Republican.
The resolution does not call for a withdrawal of troops or threaten funding of military operations, as many Democrats have suggested. Instead, it says the U.S. should transfer responsibility to the Iraqis "under an appropriately expedited timeline" that is not specified.
Biden, chairman of the Senate Foreign Relations Committee, said his panel will debate the measure on Jan. 24, the day after the president's State of the Union address. A swift committee review would pave the way for debate on the floor as early as that week, although Democrats say it is likely Republicans on the committee will want to make changes.
Biden said modest changes to the bill might be used "to attract those who share our view but may not like our specific language."
Mr. Bush sought to stave off a major showdown between the administration and Congress by inviting Republican skeptics of the plan to the White House on Wednesday. But many of those members emerged from the meeting to say they still opposed sending more troops, although they were unsure whether they would back the resolution.
Sen. Norm Coleman of Minnesota, one of several Republicans wary of Mr. Bush's plan, said he is concerned the resolution may go too far. Coleman spokesman Tom Steward said the senator is open to an increase in the Anbar province, for example.
"Senator Coleman has repeatedly conveyed his specific concerns to the president and is hopeful that Congress can find bipartisan common ground on this resolution going forward," Steward said.
Presidential politics is also part of the equation, CBS News senior White House correspondent Bill Plante reports. Many of those who spoke against the troop surge have ambitions for '08: Hagel; Sen. Sam Brownback, R-Kan.; Sen. Chris Dodd, D-Ct.; and Sen. Hillary Clinton, D-N.Y.
Alternative proposals have already begun to surface. House Republican leaders backed a bill that would protect funding for U.S. troops, while Senate Republicans prepared a resolution supportive of Mr. Bush's strategy.
Republican Sen. Lindsey Graham said that resolution would say the Senate believes the war in Iraq cannot be lost "and this strategy could bring about success if properly supported."
Sen. John Warner, a Republican, is considering an alternative proposal. Rather than denouncing the president's strategy, it would voice support for recommendations by the bipartisan Iraq Study Group. That panel did not recommend sending more troops unless specifically requested by a military commander.
Some Democrats have other plans of their own. Dodd said he wants legislation capping the number of troops in Iraq at existing levels. Clinton on Wednesday announced legislation that would require Mr. Bush to obtain congressional approval for additional troops in Iraq if the Iraqis cannot show progress after six months.
"I am opposed to this escalation," Clinton, who just returned from a trip to Iraq and Afghanistan, told .
However, Clinton added, "I am for putting more troops in Afghanistan."