By Steve Holland

WASHINGTON, May 10 (Reuters) - U.S. President George W. Bush is under mounting pressure from fellow Republicans to show substantial progress in Iraq by September or risk their desertion.

Republicans looking ahead to the 2008 U.S. presidential election, after already 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. They told him 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 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.

White House spokesman Tony Snow refused to divulge details of the meeting. Nor would he attach any particular significance to September as a time when progress needs to be evident.

Snow said Republicans are in fact united against Democratic attempts to attach a troop pullout timetable to Bush's request for $100 billion to fund the wars in Iraq and Afghanistan.

"I think if you're trying to do an unraveling (story), you don't unravel when everybody is basically on the same page in terms of what they're going to be doing with the key political decisions being made right now," Snow said.

Many military strategists have said September is too soon to reach a judgment on the effectiveness of the troop increase.

But September is looming as a critical time frame because members of Congress will have returned from August recess at home and will have heard from the voters 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 Wednesday. (Additional reporting by Peter Szekely)

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