Progress over a stalled Iraq funding bill moved one step further along after the Senate passed a key vote on Thursday, closing debate on its version of the supplemental spending bill and paving the way toward negotiations with the House.

The Senate agreed to end debate — and avoid a bill-killing filibuster — on a nonbinding resolution expressing support for troops in Iraq. The 94-1 cloture vote was followed by a voice vote formally passing the bill and sending it to a conference committee of members from both chambers.

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Those negotiations are key to coming to a final bill because the Senate was cool to the House version passed last week that would set up a two-part disbursement of the roughly $90 billion requested by the administration for ongoing military operations in Iraq. According to the House bill, Congress could decide not to give the second tranche of the money if lawmakers aren't satisfied that government benchmarks are met or security is improved. President Bush has threatened to veto the House bill.

The House-Senate negotiations — which largely are already under way informally — are closely linked to talks with White House officials in an effort to hammer out a solution for an Iraq funding bill that has been elusive since Bush first made the request earlier this year.

Shortly after the vote, House Republican leaders accused House Speaker Nancy Pelosi and Senate Majority Leader Harry Reid of foot-dragging, and called on them to speed their work to finish a bill before Memorial Day.

"I'm willing to sit down with Mrs. Pelosi, Mr. Reid and the White House to resolve these differences, and the sooner we do it, the better," House Minority Leader John Boehner said.

"Remember, Memorial Day is a national holiday to remember those who have fallen in service to our country. And for Congress to not deal with this issue to fund our troops before Memorial Day is a disservice to all of those who have given their lives to our country," said Boehner, R-Ohio.

Bush is asking for $90 billion to pay for ongoing military operations that he says are necessary to keep the troops in Iraq equipped, as well as keep up training for forces that would replace troops already deployed.

The president already has vetoed a $124.2 billion bill earlier this month, saying that it unfairly would have tied his and his commanders' abilities to fight the war because it included a timeline for withdrawal. Democrats were unable to muster enough to override the veto, and Bush has threatened to veto last week's House bill for the similar reasons.

Bush, speaking at the White House with outgoing British Prime Minister Tony Blair by his side, told reporters he believed his administration and Congress could strike a compromise.

"I think we'll get a deal. We'll work through something we can all live with," Bush said.

Sen. Russ Feingold, D-Wis., cast the lone vote against Thursday's measure. Senators who did not vote included Republicans Tom Coburn of Oklahoma, Elizabeth Dole of North Carolina, John McCain of Arizona and John Sununu of New Hampshire. Sen. Tim Johnson, D-S.D., also missed the vote while he recuperates from brain surgery he underwent in December.

Reid has said he believes a bill can be hammered out before the Memorial Day recess that begins late next week, but negotiations will be tense. Just before Thursday's scheduled vote, Senate Appropriations Committee Chairman Robert Byrd expressed his deep concerns with the Bush administration policy in Iraq.

"Our president continues to miss the point. Iraq is at war with itself. ... America cannot create a stable democracy in Iraq at the point of a gun. While our troops succeeded in toppling Saddam Hussein, it is the president's profound, profound misunderstanding of the dynamics in Iraq that has led to the failure of his Iraqi policies," said Byrd, D-W.Va.

But Byrd earlier had signaled his willingness to work.

"To be successful, we must end the finger-pointing and instead roll up our sleeves and work together. I believe that we can — and we will," Byrd said, according to The Associated Press.

Senate Minority Leader Mitch McConnell has said the "substance of a final deal" has been discussed with Reid and White House chief of staff Josh Bolten, but he did not reveal any further details.

Reid, D-Nev., told reporters that the final outcome will not give the president all he wants.

"He's not going to have the blank check," Reid said. "There's a Congress and he has to deal with us."

The contours of a deal remain fluid, but Capitol Hill sources told FOX News that the White House and Reid are moving toward a funding bill that applies no timelines for troop withdrawals but does apply non-binding benchmarks to measure Iraqi political, economic and security reforms.

Getting Pelosi on board with the approach may prove difficult, however. She has led efforts to impose troop withdrawal timelines and is so far reluctant to embrace anything less. Other House Democrats are preparing to accept a compromise, sources said.

Meanwhile, Sen. James Inhofe, R-Okla., said he is encouraged by recent developments in Iraq that appear to show the president's strategy is working.

Discussing his recent trip to Iraq with FOX News on Thursday, Inhofe said Ramadi "used to be the terrorist capital of Iraq," but the number of trained troops there has increased, and attacks on troops have decreased dramatically, by 74 percent at one count.

"It's really a success story. But if you want a real success, go down to Fallujah. That's where a lot of our viewers — I have to remind them — that's where the Marines were going door to door at one time. ... The security is now under the Iraqis in Fallujah. We have won in Fallujah," Inhofe said.

FOX News' Trish Turner and The Associated Press contributed to this report.