(Updates with Bush comments) By Steve Holland and Tabassum Zakaria WASHINGTON, May 10 (Reuters) - U.S. President George W. Bush, already under fire from Democrats over Iraq, is facing mounting pressure from fellow Republicans to show substantial progress in the war by September or risk their desertion. Republicans looking ahead to the 2008 U.S. presidential election, after losing control of the U.S. Congress in November, are publicly sharing doubts about the president's war strategy. "The American people are war-fatigued. The American people want to know that there's a way out. The American people want to know that we're having success," Illinois Republican Rep. Ray LaHood told CNN on Thursday. LaHood was among 11 moderate Republicans who met privately with Bush at the White House on Tuesday. Most, if not all, could face stiff Democratic challenges to their re-election. They told Bush that by September the troop buildup he ordered for Iraq three months ago must show progress. Particularly frustrating to members of the U.S. Congress are plans by the Iraqi parliament to take a two-month summer vacation, a subject raised by Vice President Dick Cheney on his visit to Baghdad on Wednesday. "Members really told the president, in I think the most unvarnished way that they possibly could, that things have got to change, that we're going to hang with him until September, but we need an honest assessment in September and people's patience is running very, very, very thin," LaHood said. Bush is already facing opposition to his Iraq plan -- Democrats in the U.S. House of Representatives are pushing bills that would provide partial war funding but also begin withdrawing troops. And he lost a key ally on Thursday with British Prime Minister Tony Blair's announcement he will step down in June. Bush, answering questions during a visit to the Pentagon, acknowledged some Republicans are uneasy and pleaded for more time. "They're obviously concerned about the Iraq war, but so are a lot of other people," he said. He said the U.S. commander in Iraq, Gen. David Petraeus, would report back in September on whether the new strategy is working. "What we need to give General Petraeus is plenty of time to work. This debate raging in Washington about how long we're going to be there -- we haven't even got all our troops there," he said. White House counselor Dan Bartlett said the Republicans are uneasy and added, "They reflect the sentiment of the public." Polls show a solid majority of Americans are opposed to the four-year Iraq war. Many military strategists have said September is too soon to judge the effectiveness of the troop increase. But September is looming as a critical time because members of Congress will have returned from August recess at home and will have heard from voters in their home areas about Iraq. It also marks a period in which Americans will begin to take greater interest in candidates running to succeed Bush in the November 2008 election. "When we get back in September, there's going to be a reassessment," Ohio Rep. John Boehner, the top House Republican, said on CNN on Wednesday. (Additional reporting by Peter Szekely)
