(Updates with U.S. House passing Iraq funds bill)

By Steve Holland and Richard Cowan

WASHINGTON, May 10 (Reuters) - U.S. President George W. Bush faced mounting pressure on Thursday from fellow Republicans as well as Democrats to show progress in Iraq, as Congress again defied him by passing a limited war funds bill.

On the day that Bush's closest international ally on the war, British Prime Minister Tony Blair, announced he would step down next month, Democrats in the House of Representatives passed the war funds bill he had already promised to veto.

The bill provides only enough money to continue combat for the next two or three months and lacks any guarantee of future funding.

"I'll veto the bill if it's this haphazard, piecemeal funding," Bush warned earlier in the day, saying Congress should give military commanders more time and flexibility.

Bush vetoed a $124 billion war funding bill last week because it set a deadline for the withdrawal of combat troops.

The latest Democratic-backed measure, passed by a vote of 221-205, would give Bush $42.8 billion in emergency military funds for the wars in Iraq and Afghanistan and related activities.

An additional $52.8 billion would have to be approved by Congress in late July, after Bush submits progress reports on the war. Lawmakers would then decide whether to use that second batch of money to continue combat or to bring most U.S. troops out of Iraq.

Bush wants the nearly $100 billion up front to fund the wars and without conditions for future votes on troop withdrawals.

House Appropriations Committee Chairman David Obey of Wisconsin said he and fellow Democrats had made several key concessions to Bush, including dropping earlier provisions setting firm dates for leaving Iraq.

Earlier in the day, after a meeting with U.S. commanders for an update on the war, Bush hailed Blair as a "political figure who is capable of thinking over the horizon," after the British leader made the expected announcement he would step down in June.

REPUBLICAN DOUBTS

Blair told Britons disillusioned by the war, which severely damaged both leaders' standing at home and abroad, that he had done what he believed was right.

Members of Bush's Republican Party also have begun to more publicly question the war, which has killed at least 3,377 U.S. soldiers and injured more than 24,000 since 2003.

"The American people are war-fatigued," Illinois Republican Rep. Ray LaHood told CNN on Thursday. "The American people want to know that there's a way out. The American people want to know that we're having success."

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 in 2008 elections.

They told Bush that by September the troop buildup he ordered for Iraq three months ago must show progress. But only two Republicans on Thursday voted for the funding bill passed by the House.

Bush is sending 30,000 additional troops to Iraq, mainly for the offensive in Baghdad, which is regarded as a last-ditch effort to pull Iraq back from the brink of civil war.

In a sign of growing U.S. impatience at Iraq's slowness in passing laws on oil distribution and other key measures, U.S. Vice President Dick Cheney made an unannounced visit there this week and told U.S. soldiers on Thursday that militants had made Iraq the front line in the war on terrorism.

"We are here, above all, because the terrorists who have declared war on America and other free nations have made Iraq the central front in that war," Cheney told the troops at a base near Saddam Hussein's hometown of Tikrit.

The office of Iraq's Sunni vice president, Tareq al-Hashemi, said the senior Sunni leader had been "comforted" by recent meetings with political leaders aimed at addressing Sunni Arab concerns.

Hashemi and other senior figures from Iraq's largest Sunni Arab bloc had warned they might quit the government of Shi'ite Prime Minister Nuri al-Maliki if their grievances were ignored.

On Wednesday, Cheney held talks with Hashemi, Maliki and President Jalal Talabani, a Kurd.

Sunnis want to increase their representation in Iraq's security forces and soften laws that prohibit thousands of members of Saddam's former Sunni-dominated Baath Party from playing a role in public life.

Democratic lawmakers argue that withholding some funds could put pressure on Iraqis to step up attempts to stabilize their country. (Additional reporting by Dean Yates and Mussab Al-Khairalla, and Richard Cowan in Washington)

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