WASHINGTON - President Bush and congressional leaders began negotiating a second war-funding bill yesterday, with Democrats offering the first major concession: an agreement to drop their demand for a timeline to bring troops home from Iraq.

Democrats backed off after the House failed, on a vote of 222-203, to override the president's veto of a $124 billion measure that would have required U.S. forces to begin withdrawing as early as July. All area representatives voted with their party.

The House vote was 62 shy of the two-thirds majority needed to override a veto. With few exceptions, Republicans stood fast with Bush in the wartime clash.

But Democratic leaders made clear that the next bill would have to include language that influences war policy.

Senate Majority Leader Harry Reid (D., Nev.) outlined a second bill that would step up Iraqi accountability, "transition" the U.S. military role, and show "a reasonable way to end this war."

"We made our position clear," House Speaker Nancy Pelosi (D., Calif.) said after a White House meeting with the president. "He made his position clear. Now it is time for us to try to work together.

"But make no mistake," she said. "Democrats are committed to ending this war."

Bush said he was "confident that we can reach agreement" and assigned three top aides to negotiate. White House chief of staff Joshua Bolten, national security adviser Stephen Hadley, and budget director Rob Portman will go to Capitol Hill today to sit down with leaders of both parties.

But a new dynamic also is at work, with some Republicans now saying that funding further military operations in Iraq with no strings attached does not make practical or political sense. Rep. Bob Inglis (R., S.C.), a conservative who opposed the first funding bill, said, "The hallway talk is very different from the podium talk."

In yesterday's House vote, 220 Democrats and two Republicans voted to override the president's veto, and 196 Republicans and seven Democrats voted to sustain it.

While deadlines for troop withdrawals had to be dropped from the spending bill, such language is likely to appear in a defense-policy bill that is expected to reach the House floor in two weeks, just when a second war-funding bill could be ready for a House vote. Democrats want the next funding bill to pass before Congress recesses May 25 for Memorial Day weekend.

Beyond that, Democrats remain deeply divided over how far to give in to the White House.

House Majority Leader Steny H. Hoyer (D., Md.) indicated that the next bill would include benchmarks to keep the Iraqi government on course, such as passing a law to share oil revenue, quelling religious violence, and disarming sectarian militias.

Failure to meet benchmarks could cost the Iraqi government billions of dollars in nonmilitary aid, and the administration would be required to report to Congress every 30 days on the military and political situation in Iraq.

Benchmarks have emerged as the most likely foundation for bipartisan consensus and were part of yesterday's White House meeting, participants said.

"I believe the president is open to a discussion on benchmarks," said Senate Democratic Whip Richard J. Durbin of Illinois, who attended the White House session.

He added that no terms were discussed. "We didn't go into any kind of detail," Durbin said.

A growing number of Republican lawmakers want language that would hold the administration and the Iraqi government more accountable.

"The general sense is that the benchmarks are critical," said Sen. Olympia Snowe (R., Maine), a moderate who opposed the original bill but supports some constraints.

White House officials are also looking to benchmarks as an area of compromise, but they want them to be tied to rewards for achievement, not penalties for failure.

Administration officials note that they do not oppose benchmarks and in fact have developed them in the past along with Iraqis.

But that approach would be too weak even for moderates from both parties. Already, liberal Democrats believe that public opinion and circumstances in Iraq are on their side, and view benchmarks alone as far too weak.

Conservative Republicans were balky.

Sen. Ted Stevens of Alaska argued that any benchmarks would make the bill "unconstitutional."

Sen. Lindsey Graham of South Carolina, another unwavering Bush defender on Iraq, said he would support adding benchmarks, but with no repercussions should the Iraqis fall short.