WASHINGTON — On the eve of a critical vote, House Democrats labored Thursday to lock down a majority behind a Sept. 1, 2008, deadline for the withdrawal of U.S. combat troops from Iraq, the sternest test yet for a determined new majority eager to challenge President Bush.

“If it comes off, it’s a superb accomplishment,” said Rep. Barney Frank, D-Mass., as the party’s leaders cajoled liberals who want an even faster timetable and moderates fearful of tying the hands of the commander in chief and generals in the field.

Democratic aides expressed growing confidence of success when the vote is called, and four of the bill’s most consistent critics said they had told Speaker Nancy Pelosi they would help pass it, even though they intend to personally vote against it because it is not strong enough.

“While I cannot betray my conscience, I cannot stand in the way of passing a measure that puts a concrete end date on this unnecessary war,” said one of the four, Rep. Barbara Lee of California. With the leadership lobbying intensively on its own, it was not clear which lawmakers, if any, had swung behind the bill as a result of the offer the four had made.

Throughout the day, a string of liberal opponents of the war swung behind a measure they deemed insufficient.

“I want this war ended today. If I thought it would help this war ending sooner by voting against the bill, I would vote against it in a heartbeat,” said Rep. Jim McGovern of Massachusetts, who sponsored legislation for a troop withdrawal in 2005.

“But I don’t believe that to be the case,” he added of the measure, which combines funding for the war, the troop withdrawal deadline and billions of dollars in funding for politically popular programs at home ranging from farm aid to relief for the victims of Hurricane Katrina.

The legislation marks Congress’ most direct challenge to date of Bush’s policy in a war that has claimed the lives of nearly 3,200 U.S. troops.

As debate began in the House, Republicans criticized it vociferously. “The bill is a sham,” said Rep. Harold Rogers of Kentucky, adding it would “provide fodder for our enemies abroad.”

Bush has threatened to veto the bill, opposing both the troop withdrawal provision and billions of dollars in spending that Democrats added.

Defense Secretary Robert Gates said any delay in approving the funds could “have a genuinely adverse affect on the readiness of the Army and the quality of life for soldiers and their families.”

White House press secretary Tony Snow sharpened the message. “There’s a very real chance that money for the troops will run out while members of Congress are on vacation,” he said. “Is that the message you want to send to men and women who are putting their lives on the line?”

Democrats hold 233 seats in the House, meaning they can lose 15 votes from their own rank and file and still be assured of passing the measure.

Whatever the vote count, some war foes disagreed with the strategy.

“We don’t agree with them,” said Rep. Lynn Woolsey, D-Calif. She said she and others believe the party must “honor what the voters of November said, which is to be bold, end the war and bring the troops home.”