WASHINGTON (CNN) -- There were fresh signs Wednesday that some Republican senators are impatient with the White House plan for Iraq and are demanding that the president change his war strategy now.
After emerging from an hourlong private meeting in the Capitol in which they had a "vigorous" exchange with National Security Adviser Stephen Hadley, several senators said they urged the president to begin planning now for a repositioning of U.S. involvement in Iraq after the American commander there, Gen. David Petraeus, releases a progress report in September.
Bush said Tuesday he would not alter his current war plan until after September, when his top U.S. general in Iraq is scheduled to reveal whether the current troop "surge" is working.
But the president is fighting a tide of dissent among lawmakers in his own party, while Bush and the war itself rank low in national opinion polls. Watch White House dodge Iraq questions »
Sen. Norm Coleman of Minnesota, who is up for re-election in 2008 and is facing tough criticism for his continued support of the Iraq war, said after the meeting with Hadley that "I don't think any votes or minds were changed in there."
"I offered concern about the Iraqi government," Coleman said. "I think it's important to have a bipartisan approach. The principles are clear -- there's got to be a long-term presence [in Iraq] but a change of mission."
But Sen. Robert Bennett of Utah said he thought the senators and Hadley had a "very useful exchange," which he described as "vigorous."
"He told us where the president is, and we told him where we are, and we reached towards each other," Bennett said.
Not all the Republicans in the meeting were critical of the Bush policy. Sen. Jim DeMint of South Carolina said that "we told Petraeus that we'd send our troops there and gear up, and for us to even be debating a change before September 15 is irresponsible government."
The uneasiness comes as the administration nears release of a report on 18 benchmarks for the Iraqi government. Two senior White House sources told CNN the report may be released as early as Thursday.
Republicans voting for cloture:
Sen. Norm Coleman, Minnesota
Sen. Susan Collins, Maine
Sen. Chuck Hagel, Nebraska
Sen. Gordon Smith, Oregon
Sen. Olympia Snowe, Maine
Sen. John Sununu, New Hampshire
Sen. John Warner, Virginia
Independents voting against cloture:
Sen. Joseph Lieberman, Connecticut
Senators not voting:
Sen. Sam Brownback, R-Kansas
Sen. Tim Johnson, D-South Dakota
Sen. David Vitter, R-Louisiana
The 23-page classified report will conclude the Iraqi government is making "satisfactory" progress on about half of the benchmarks identified by Congress but "unsatisfactory" progress on the rest, including political reform and a plan for sharing oil revenue, sources familiar with the report told CNN on Wednesday. Many of the milestones are aimed at quelling Iraqi sectarian violence.
After a meeting at the White House Wednesday, Sen. John McCain, R-Arizona, said the report will show military success but "on the political side, there is none."
Three Republican senators -- Sens. Bennett, Lamar Alexander of Tennessee, and Pete Domenici of New Mexico -- agreed to back a bipartisan amendment that would implement the recommendations of the Iraq Study Group, which was headed by former Republican Secretary of State Jim Baker and former Democratic Rep. Lee Hamilton.
The amendment is authorized by Sen. Ken Salazar, D-Colorado.
"Basically, the White House position is we should wait to hear from Petraeus. My view is that we should adopt Baker-Hamilton recommendations now," said Alexander. "I think [President Bush] should do that today."
In December, the Iraq Study Group recommended that 25 percent of the U.S. combat troops be withdrawn from Iraq by the first quarter of 2008 and said the remaining troops should be shifted to more secure positions.
Bush instead sent 30,000 more U.S. troops into Iraq to try to control the violence there.
Implementing the Baker-Hamilton recommendations would allow for redeployment of American forces by next spring, but without mandating a withdrawal, as some Democratic lawmakers have proposed.
The three GOP senators' decision to back the Salazar amendment comes as the president has suffered numerous Republican defections from his position on the war his week.
Republican Sens. Chuck Hagel and Olympia Snowe added their names as co-sponsors of a Democratic measure for U.S. forces to pull out of Iraq by April. They joined a third GOP sponsor of the legislation, Sen. Gordon Smith of Oregon.
Also, a new bill by powerful Republican Sens. John Warner, the former ranking Republican on the Senate Armed Services Committee, and Richard Lugar, the top Republican on the Senate Foreign Relations Committee, would tie U.S. policy in the war to last year's Iraq Study Group report, according to the Washington Post.
Meanwhile, in a significant measurement of GOP loyalty to the president's Iraq policy, a Democratic amendment to limit troop deployment failed to advance Wednesday, providing a victory for the White House.
The tally was 56 to 41, failing to receive the 60 votes required to cut off debate on the amendment.
The cloture vote Wednesday was on a proposal by Jim Webb, D-Virginia, that would have forced military personnel who return from deployments to remain stateside for at least as long as they spent overseas before they could be sent back to war. E-mail to a friend
CNN's Dana Bash, Evan Glass, Ed Henry and Charley Keyes contributed to this report.