Even in Sunni-dominated Anbar Province, where insurgents are carrying out a vicious guerrilla war against foreign troops, a handful of leaders are asking American commanders to rein in Iraqi paramilitary units. Sheiks in Falluja often complain to American officers there of harassment, raids or indiscriminate shooting by Iraqi forces.

A year ago, the party of Tariq al-Hashemi, a hard-line Sunni Arab who is one of Iraq’s two vice presidents, was calling for the immediate withdrawal of foreign troops.

“The situation is different now,” Mr. Hashemi said. “I don’t want the Americans to say bye-bye. Tomorrow, if they were to leave the country, there would be a security vacuum, and that would lead inevitably to civil war.”

Zalmay Khalilzad, the American ambassador to Iraq, has been at the forefront of American efforts to bring Sunni Arabs into the political process. Part of that strategy is to crack down on Shiite militias and push for amnesty for some guerrillas.

This month the American military has stepped up operations against the Mahdi Army, a volatile Shiite militia, and the top American commander, Gen. George W. Casey Jr., said last Wednesday that the Americans would hunt down “death squads” that are a driving force behind the rising bloodshed.

Some Shiite leaders deride the American policy toward Sunnis as appeasement. “This strategy will destroy their goal of establishing democracy in Iraq,” said Abbas al-Bayati, a prominent Shiite legislator. “Compromising with the insurgency will encourage the insurgents to do more and more violence in the region.”

Investigations into possible wrongdoing by American troops in two major cases — the deaths of 24 civilians in Haditha last November, and the rape and murder of a 14-year-old girl and the killing of her family in Mahmudiya in March — have ignited anger among Sunnis, but not nearly to the same degree as they might have in 2004, when the Abu Ghraib prisoner scandal emerged. But back then, Iraq had not crept to the brink of full-scale civil war.