With various groups working on the sidelines to craft an amendment on the Iraq War that can garner bipartisan support in the Senate, Majority Leader Harry Reid (D-Nev.) attempted to step up the pressure on Republicans on Tuesday by accusing them of blocking progress on the Defense Department authorization bill.

“It’s hard for me to comprehend how they’re going to filibuster the Defense authorization bill, but that’s what they’re going to do,” Reid said at a press conference.

Meanwhile, two Senate Republicans made noises that they may support amendments that force the president to change his strategy in Iraq, even as the White House issued a veto threat if the bill “sets an arbitrary date for beginning the withdrawal of American troops.”

Sen. George Voinovich (R-Ohio), who has recently been critical of the administration’s war strategy, said he did not know how he would vote on a number of proposals to begin redeploying U.S. troops out of Iraq, but he expressed interest in putting pressure on the White House to begin extricating the United States from the war-torn country.

“I think we should continue to ratchet up the pressure, in addition to our own words, to let the White House know we’re serious about this,” Voinovich said. “I’m not telling you whether I’m going to vote or not going to vote” for any specific amendment.

Voinovich also expressed skepticism about whether an amendment designed to implement the recommendations of last year’s bipartisan Iraq Study Group would be forceful enough.

“The question I have is, will that be enough of an incentive for the administration to move to a more comprehensive disengagement strategy,” he said.

Sen. Olympia Snowe (R-Maine) also reiterated her interest in voting for a proposal — to be offered by Armed Services Chairman Carl Levin (D-Mich.) and Sen. Jack Reed (D-R.I.) — that would force the president to withdraw combat troops, but not support and counterterrorism troops, from Iraq by April 30, 2008.

Reid’s tack on Tuesday appeared to be a dual attempt to press Republicans on the political question of Iraq and to challenge the need for 60 votes for passage of any measure — one of Minority Leader Mitch McConnell’s (R-Ky.) chief tactical tools this year.

In fact, Reid and McConnell traded barbs on the floor over the process for debating the bill, with Reid saying McConnell’s insistence on pre-setting 60-vote thresholds for all Iraq-related amendments amounted to a defacto filibuster. Sixty votes are required to cut off a filibuster.

McConnell argued “every Iraq amendment we’ve voted on this year … as most things in the Senate that are remotely controversial, has required 60 votes.” McConnell added that his suggestion to only adopt amendments that receive 60 or more votes was “the rational response to the nature of the Senate in this era.”

Reid fired back that “rationality is in the eye of the beholder. That’s just some recent rule that’s come up in the minds of the minority.”

At issue was an amendment by Sen. Jim Webb (D-Va.) to require troop rotations to adhere to a formula by which soldiers spend at least as much time at home as they’ve spent deployed to war zones before they can be redeployed. National Guard and Reserve forces would have to be given three times as much time at home as they are deployed.

Webb has said the amendment would address the administration’s decision to increase the amount of time soldiers spend deployed as well as the Pentagon’s decision to send troops back for multiple tours of duty without what he says is sufficient time at home.

But Republicans, such as Sen. Lindsey Graham (S.C.), argue the amendment “would unravel the surge” strategy the president began earlier this year by forcing the military to begin bringing soldiers home. “It’s a not so clever effort to do in the backdoor what you couldn’t do in the front door,” Graham said.

Reid offered to have simple majority votes on the Webb amendment and a Graham proposal that would state the sense of the Senate that soldiers should be given adequate rest before being redeployed. When McConnell objected, Reid filed a cloture motion to limit debate on the Webb amendment.

A vote on that motion is expected to occur this morning. If Webb does not garner the 60 votes necessary for his amendment, it is unclear what Democrats’ next move may be.

Levin said Tuesday that Reid may decide to stay on the amendment until it can be “disposed of,” which could indefinite debate. Additionally, Republicans could object to moving to any other amendments — a process that would require unanimous consent.

If the Senate votes to limit debate on the Webb amendment, Reid has indicated he will bring up the Levin-Reed amendment to force the Pentagon to begin reducing the number of troops in Iraq within 120 days of enactment and to change the mission in Iraq from combat to support, training and counterterrorism. The Levin-Reed amendment does not specify how many troops would have to be removed.

That amendment also could face a filibuster, and the White House on Tuesday officially threatened to veto the entire bill if it is included.

But several centrists were still looking for a middle ground on Iraq. Sens. Susan Collins (R-Maine) and Ben Nelson (D-Neb.) have been drafting an amendment that would redefine the U.S. mission in Iraq but would not set any specific dates for a drawdown of troops. Nelson said it was a “realistic, tactically feasible plan.”

Collins added that she is also in talks with Sens. John Warner (R-Va.) and Dick Lugar (R-Ind.). Though Warner is working on a proposal, he has said he would not offer any amendments on Iraq until after the White House gives a report expected this week on the status of the president’s “surge” strategy to secure Baghdad.

And Sens. Ken Salazar (D-Colo.) and Lamar Alexander (R-Tenn.) were still trying to build support for their amendment to establish the Iraq Study Group’s recommendations as official U.S. policy, but many Democrats derided the amendment, which would not set any firm timelines for withdrawal, as too weak.

“I’d probably vote for Salazar. It’s a harmless amendment,” Levin said.

Following a leadership meeting Tuesday evening, House Democrats said they will bring to the floor, likely Thursday, stand-alone legislation that would begin the redeployment of military personnel from Iraq within 120 days of the bill’s passage and would require that withdrawal to be completed by April 1, 2008.

“We want to keep momentum to change the course in Iraq,” said Kristie Greco, spokeswoman for Majority Whip James Clyburn (D-S.C.) who expressed confidence the legislation would pass. Aides said Rep. John Tanner (D-Tenn.), a prominent Blue Dog, is backing the effort. The Caucus may meet today to discuss the proposal, and leadership was working at press time to circulate draft legislation to its members.

Minority Whip Roy Blunt (R-Mo.) said House Republicans would stick together on Iraq until September, at which point it was “likely” the mission in Iraq would have to “evolve.”

Meanwhile, the White House’s veto threat said advisers would recommend a veto if the bill retains language giving terrorism detainees more legal avenues to protest their detention, if it retains a proposal barring the White House from requiring intelligence offers to get approval before submitting testimony to Congress, and if it retains language requiring the White House to turn over intelligence reports requested by the Senate Armed Services Committee.

Notably, the White House did not threaten a veto if the Webb amendment were adopted.

John Stanton and Susan Davis contributed to this report.