The move puts the prominent Republican at odds with the president, who says conditions on the ground should dictate deployments.
Warner, R-Va., said the troop withdrawals are needed because Iraqi leaders have failed to make substantial political progress, despite an influx of U.S. troops initiated by Bush earlier this year.
Warner says the departure of even a small number of U.S. service members — perhaps 5,000 out of the 160,000 troops in Iraq — would send a powerful message throughout the region that time is running out.
"We simply cannot as a nation stand and continue to put our troops at continuous risk of loss of life and limb without beginning to take some decisive action," he told reporters after a White House meeting with top aides to President Bush.
Sen. Warner – just back from Iraq – said U.S. soldiers are now fighting for a failing government, reports CBS News correspondent Bob Orr.
"I really firmly believe the Iraqi government under the leadership of Prime Minister Maliki has let our troops down," Warner said.
It's the messenger, not the message, that is important, says CBS News chief Washington correspondent Bob Schieffer.
"John Warner is the single most influential Republican voice on Capitol Hill," says Schieffer. "Other Republicans listen when he's talking about defense matters."
Warner's new position is a sharp challenge to a wartime president that will undoubtedly color the upcoming Iraq debate on Capitol Hill. Next month, Gen. David Petraeus, the top military commander in Iraq, and U.S. Ambassador Ryan Crocker are expected to brief members on the war's progress.
A White House spokesman, Gordon Johndroe, declined to say whether President Bush might consider Warner's suggestion.
Asked whether President Bush would leave the door open to setting a timetable, Johndroe said: "I don't think the president feels any differently about setting a specific timetable for withdrawal. I just think it's important that we wait right now to hear from our commanders on the ground about the way ahead."
Republicans, including Warner, have so far stuck with Mr. Bush and rejected Democratic proposals demanding troops leave Iraq by a certain date. But an increasing number of GOP members have said they are uneasy about the war and want to see President Bush embrace a new strategy if substantial progress is not made by September.
Warner, known for his party loyalty, said he still opposes setting a fixed timetable on the war or forcing the president's hand.
"Let the president establish the timetable for withdrawal, not the Congress," he said.
Nevertheless, his suggestion of troop withdrawals is likely to embolden Democrats and rile some of his GOP colleagues, who insist lawmakers must wait until Petraeus testifies.
His stature on military issues also could sway some Republicans who have been reluctant to challenge a wartime president. Warner is a former Navy secretary and one-time chairman of the Senate Armed Services Committee; he is now the committee's second-ranking Republican.
Warner said he came to his conclusion after visiting Iraq this month with Democratic Sen. Carl Levin of Michigan, committee chairman. Earlier this week, Levin said Prime Minister Nouri al-Maliki should be replaced. Sen. Dianne Feinstein, D-Calif., followed suit, saying Maliki has been "a failure."
Warner said he "could not go that far" to call for Maliki's resignation but said he did have serious concerns about the effectiveness of the current leadership, confirmed by an . The National Intelligence Estimate on Iraq predicted it would be 12 months before the U.S. could expect a reconciliation.
"When I see an NIE which corroborates my own judgment — that political reconciliation has not taken place — the Maliki government has let down the U.S. forces and, to an extent, his own Iraqi forces," he said.
Senate Majority Leader Harry Reid said the report confirms what most Americans already know: "Our troops are mired in an Iraqi civil war and the president's escalation strategy has failed to produce the political results he promised to our troops and the American people."
"Every day that we continue to stick to the president's flawed strategy is a day that America is not as secure as it could be," said Reid, a Nevada Democrat.