Opening the latest round of debate on the Iraq War, the Senate could vote as early as Tuesday on a measure that would mandate longer intervals between troop deployments.

The amendment to the defense authorization bill (HR 1585, S 1547), by Jim Webb, D-Va., and Chuck Hagel, R-Neb., is one of several that senators plan to offer that would redefine or end the U.S. mission there.

These latest congressional challenges to President Bush come as the White House prepares to report to Congress by week’s end on progress and setbacks in Iraq.

As the debate begins, the Congressional Research Service reported Monday that the average monthly cost of the conflicts in Iraq and Afghanistan had been clocked at $12 billion. Two years ago, the CRS put the monthly figure at $7.7 billion.

The debate also comes as GOP support for Bush’s war strategy is weakening. In recent weeks, several senior Senate Republicans — including Richard G. Lugar of Indiana, George V. Voinovich of Ohio, and Pete V. Domenici of New Mexico — have broken with the White House on Iraq and urged Bush to begin redeploying U.S. forces.

“A growing number of Republicans are now speaking against the failed strategy in Iraq, and that’s good,” said Majority Leader Harry Reid, D-Nev. “And these Republican defections are apparently leading the White House to consider changing its mission. That’s good.”

The test, Reid said, is “for those Senate Republicans who are saying the right things on Iraq, they must put their words into action by voting with us to change course and responsibly end this war.”

None of the Republicans, however, say they are prepared to vote for a fixed timetable to begin or complete a U.S. withdrawal.

Webb’s amendment is expected to be the first of those test votes. It would require soldiers to spend as much time at home between deployments to Iraq or Afghanistan as they do fighting there.

Webb said he hoped to halt the disastrous effect that frequent combat tours are having on U.S. military readiness. “Our ground forces in particular are being burned out,” he said.

The amendment also would allow the Pentagon to send National Guard and Army Reserve troops back to Iraq or Afghanistan only three years after their last combat tour.

It would allow the president to waive the requirement for a minimum rest time at home if he can certify to Congress that deploying that unit or person “is necessary to meet an operational emergency posing a threat to vital national security interests of the United States.”

Senate critics said the threshold for that waiver authority was too high and that the measure would limit the commander in chief’s flexibility in wartime. “I’m very uneasy about the Congress dictating the deployment policy of troops,” said Republican Jeff Sessions of Alabama.

Lindsey Graham, R-S.C., plans to offer an alternative, non-binding amendment that would express the Senate’s sense that reservists should not be deployed for more than one year out of their six years of service and that active-duty forces have a year at home between deployments.

On Monday evening, Senate leaders were unable to reach agreement on a schedule for debating the Webb amendment. Reid suggested he might file a cloture motion Tuesday in an effort to shut off debate if no agreement is reached by then.

Armed Services Chairman Carl Levin, D-Mich., is expected to introduce an amendment on Tuesday that would require U.S. forces to begin their withdrawal from Iraq in roughly 120 days from enactment of the defense measure, with a waivable goal of completing the withdrawal in another 180 days.

Susan Collins, R-Maine, and Ben Nelson, D-Neb., also were crafting a measure that would redefine the U.S. mission in Iraq. “We should be planning for a significant drawdown of our forces next year,” Collins said. “A change in the mission should be as binding as you can make it.”

John W. Warner, R-Va., who has been speaking with Nelson, Collins and others about legislative options, said he would withhold comment on his proposals until after Bush gives Congress and the nation his latest update on the war.

Bart Jansen and Adam Graham-Silverman contributed to this story.