“I walked away from it feeling I got a chance to make my points,” Mr. Davis said.

The delegation included Representatives Mark Kirk of Illinois, another leader of the moderate coalition; Jim Gerlach of Pennsylvania; James T. Walsh of New York; and Jo Ann Emerson of Missouri. Mr. Kirk, Mr. Walsh and Ms. Emerson declined to discuss the meeting.

White House officials said Mr. Bush welcomed the observations of the lawmakers. “The president encouraged the members to give unvarnished opinions and views,” said Dana Perino, a White House spokeswoman. She also noted a “persistent push” by the administration in recent days to put new pressure on the Iraqi government via a secure video conference by Mr. Bush with Prime Minister Nuri Kamal al-Maliki and the surprise visit to Iraq by Vice President Dick Cheney.

The White House on Wednesday promised a veto of the emerging House bill, which would essentially provide financing for combat operations through midsummer, but require the president to provide a series of reports on the state of the Iraqi military and the progress of the government in achieving political unity. Congress would then vote a second time in late July on releasing the rest of the money sought by the administration, or restricting its use to redeployment and more limited operations in Iraq.

Tony Snow, the White House spokesman, said White House officials, led by Chief of Staff Joshua B. Bolten, would try to reach a compromise with Congress. Mr. Bolten met Wednesday with Senate leaders.

While the Pentagon awaits the money, Defense Secretary Robert M. Gates told a Senate committee on Wednesday that the military continued to shift funds, terminate contracts and slow spending so troops in Iraq and Afghanistan did not run out of money. The cost-cutting measures could sustain the troops until July, he said, “if we pulled out all the stops.”

Mr. Gates, who also attended the White House meeting on Tuesday, told lawmakers that the Pentagon would evaluate the violence in Iraq and the progress of the administration’s troop buildup plan by early September to determine the next phase of the military strategy.

“I think if we see some very positive progress and it looks like things are headed in the right direction,” Mr. Gates said, “then that’s the point at which I think we can begin to consider reducing some of these forces.”