WASHINGTON —  The Senate Foreign Relations Committee chairman on Sunday dismissed criticism a resolution opposing a troop buildup in Iraq would embolden the enemy and estimated perhaps only 20 senators believe President Bush "is headed in the right direction."

"It's not the American people or the U.S. Congress who are emboldening the enemy," said Sen. Joe Biden, D-Del., and White House hopeful in 2008. "It's the failed policy of this president — going to war without a strategy, going to war prematurely."

The Senate's top Republican, Mitch McConnell, cast doubt that a clear majority would be able to coalesce behind one of the many competing resolutions on Iraq. "I'm not certain any" will get the necessary votes, he said.

The Democratic-controlled Senate plans to begin debate this week on a nonbinding resolution declaring that Bush's proposal to send 21,500 more troops to Baghdad and Anbar province is "not in the national interest."

Last week, Biden's committee approved the measure on a near party-line vote of 12-9.

In reaction, Bush challenged lawmakers not to prematurely condemn his buildup and Vice President Dick Cheney said the administration would proceed even if a nonbinding resolution won Senate approval.

With the Senate having just confirmed a new top U.S. commander for Iraq, Defense Secretary Robert Gates said it was "pretty clear that a resolution that in effect says that the general going out to take command of the arena shouldn't have the resources he thinks he needs to be successful certainly emboldens the enemy and our adversaries."

Senate Republicans mostly oppose the committee-passed measure. They are lining up alternatives that express concern about a buildup or in other cases set performance benchmarks for the Iraqi government.

McConnell said Republican leaders would not seek to block a vote on the nonbinding resolution with a filibuster. He called a proposed resolution that focuses on benchmarks "the best way to go."

"I think I can pretty well speak for virtually all Republican senators when I say this is the last chance for the Iraqis to step up and do their part," said McConnell, R-Ky.

Biden acknowledged that votes in Congress could splinter among several competing proposals but contended that Senate opposition to the buildup was widespread.

"We will have a full throated debate on this policy," Biden said. "I will make you a bet, you will not find 20 percent of the Senate standing up and saying the president is headed in the right direction."

Cheney said most Republicans "recognize that what's ultimately going to count here isn't sort of all the hurrah that surrounds these proposals so much as it's what happens on the ground on Iraq. And we're not going to know that for a while yet," according to a Newsweek interview released Sunday.

Cheney again cited "significant progress" in Iraq and said the war is part of a long-term fight against extreme elements of Islam.

"It's not something that's going to end decisively, and there's not going to be a day when we can, say, 'There, now we have a treaty, problem solved,"' Cheney said. "It's a problem that I think will occupy our successors maybe for two or three or four administrations to come."

Sen. Richard Lugar, the top Republican on the Foreign Relations committee, said the public's concern against the war was evident by the tens of thousands of demonstrators who turned out for a protest rally Saturday in Washington.

But he said a congressional resolution would not be constructive, expressing optimism that Lt. Gen. David Petraeus, the new U.S. commander in Iraq, can do a capable job.

"I don't believe that it's helpful right now to show there's disarray around the world as well as in our body at home," said Lugar, R-Ind. "We really need, at this point, to get on the same page."

Biden and Lugar appeared on ABC's "This Week" and McConnell spoke on "Face the Nation" on CBS.