The turnabout in Mr Bush’s position came as Republicans expressed anxieties about the war and the House was expected to pass legislation late last night that would cut off funding for US troops as early as July.
Mr Bush said he would veto the measure.
“We reject that idea. It won’t work,” the president said, speaking to reporters at the Pentagon after a briefing on Iraq and Afghanistan.
Last night’s bill was opposed by nearly all Republicans and unlikely to survive in the Senate.
But House Democratic leaders say the measure shows they refuse to back down in challenging Mr Bush on a deeply unpopular and costly war.
Senate majority leader, Harry Reid, told reporters he had felt a shift in the administration’s approach to Democrats.
“It’s very clear that the people around the president recognise there are some problems,” said Mr Reid.
“And I think I have felt with my conversations with administration officials that there is a right admission that things are not going very well.”
Senate Democrats said they anticipated a vote on a war bill by next week, although Mr Reid said it remained unclear what the Senate bill might look like.
“There are 150 scenarios as to how this matter is going to be handled,” said Mr Reid. Finding a bill the Congress would pass and the president would sign was “extremely difficult”, he added.
Mr Bush pressured Iraqi leaders to move swiftly on a number of long-pending measures, including legislation to share Iraq’s oil wealth, hold provincial elections and update the constitution.
“They have got to speed up their clock,” the president said.
Washington is unhappy that Iraq’s parliament plans to take a two-month vacation this summer in the midst of the war.
Mr Bush’s willingness to put benchmarks in a war-funding bill represented a shift by the president.
“One message I have heard from people of both parties is that benchmarks make sense, and I agree,” Mr Bush said.
He said his chief of staff, Joshua Bolten, would talk with congressional leaders “to find common ground” on benchmarks.
White House officials decided Mr Bush, after refusing to discuss his negotiating stand, should declare what he was for since he had been emphatic about what he was against.
The Democrats’ bill in the House would provide the military with $42.8 billion to keep operations going through July, buy equipment and train Iraqi and Afghan security forces. Congress would decide shortly before its August recess whether to release an extra $52.8bn for war spending through September.