(Refiled to add dropped word "a" in paragraph 6)

By Daniel Bases

NEW YORK, Sept 27 (Reuters) - Ankara, an important ally for Washington in the Muslim world, would consider allowing the U.S. military to withdraw from Iraq through Turkey, Turkish Prime Minister Tayyip Erdogan said on Thursday.

Turkish-U.S. relations were strained by Turkey's refusal to allow the United States to use its territory to stage the 2003 invasion of Iraq.

Ties have been further strained by Ankara's charge that the U.S. military has done too little to keep Kurdish militants hiding in northern Iraq from attacking Turkey.

But asked on Thursday if he would allow the United States to use Turkish territory for a withdrawal from Iraq, Erdogan said through an interpreter: "We would look at this positively. We would of course have to assess the situation as to the modalities of how that would have to work."

Erdogan, whose comments came after a speech to the Council on Foreign Relations, also said Washington should set a timeline for withdrawal from Iraq.

U.S. Defense Secretary Robert Gates rejected Erdogan's call for a date certain to leave Iraq and would not specifically discuss the possibility of withdrawing through Turkey.

"I would defer to our logisticians on that," he told Pentagon reporters. "I mean, it's always nice to have somebody willing to work with you."

The withdrawal of U.S. troops has become a hot-button issue ahead of the November 2008 presidential election.

President George W. Bush approved a plan from the top U.S. commander in Iraq to start slowly reducing troop levels this month. The reduction should bring the U.S. force, now at 165,000, down by 20,000 to 30,000 by mid-2008.

Some Democrats have called for a faster withdrawal.

(Additional reporting by Kristin Roberts in Washington)

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