With less than 60 days before the complete withdrawal of U.S. forces from Iraq, the United States is quietly seeking to keep custody of its highest profile detainee there, U.S. and Iraqi officials say.

But it is unclear if Baghdad will agree -- something it appears highly reluctant to do -- or where the United States would take him if it did win outright custody.

Hezbollah operative Ali Mussa Daqduq, suspected of orchestrating a 2007 kidnapping that resulted in the killing of five U.S. military personnel, must be transferred to Iraqi custody by the end of this year under the terms of a U.S.-Iraq security agreement.

But U.S. lawmakers fear that Iraq will be unable to hold Daqduq, who was born in Lebanon, for long. Two U.S. officials, speaking on condition of anonymity, said the Obama administration would like for the Iraqis to release him to U.S. custody.

Daqduq's fate and the difficult question of what the U.S. government would do with him highlights one of the big dilemmas facing President Barack Obama as he moves forward with plans announced last month for a complete military pullout from Iraq.

Although violence has ebbed considerably since the height of Iraq's sectarian slaughter, questions remain about Iraq's ability to deal with militants -- including Daqduq, who is accused of training Iraqi extremists in how to use mortars, rockets and explosively formed penetrators, known as EFPs.

One of the U.S. officials said it was unclear whether a formal request had been made to transfer Daqduq to U.S. custody. But two Iraqi sources, including a senior Iraqi military official, said the United States had already asked to take him out of Iraq.

They asked to take him but the Iraqis are rejecting that, said the Iraqi military official, who asked not to be named because he was not authorized to speak on the record.

A senior Obama administration official acknowledged serious and ongoing deliberations about how to handle Daqduq but would not elaborate. The Pentagon declined comment.

GUANTANAMO BAY?

Daqduq was captured in March 2007 and initially claimed he was a deaf mute. U.S. forces accused him of being a surrogate for Iran's elite Quds force operatives and say he joined the Lebanese Hezbollah in 1983.

The senior Iraqi officer said he is being held in a prison jointly run by the United States and Iraq, and also cited efforts by individuals in Lebanon and Iran to win custody.

The Iranians and the Lebanese are trying to get him back by negotiating with the Iraqi government, the officer said, without elaborating.

It's unclear what options Obama would pursue to deal with Daqduq if he were taken out of Iraq. It appears highly unlikely Obama would want to add to the population at the Guantanamo Bay detention facility in Cuba, which he promised but has so far failed to close.

Robert Chesney, a counter-terrorism law and policy expert at the University of Texas School of Law, said the Daqduq case highlights the legal quandary Obama faces in detainee policy.

Although Daqduq could theoretically face trial in a U.S. civilian court, a U.S. military commission like the one in Guantanamo Bay would appear to be the most appropriate given his alleged battlefield activities. But whichever option the administration chooses, there will be fierce criticism.

If you do it one way, the left gets mad, if you do it another way, the right gets mad, Chesney said.

Senator John McCain, Senate Republican leader Mitch McConnell and 18 other lawmakers wrote to Defence Secretary Leon Panetta in July saying that bringing Daqduq to Guantanamo would be the best decision.

It is absolutely clear that the policy option that most reduces the risk to Americans' safety is the one the administration apparently refuses to consider -- law of war detention at Guantanamo with or without trial by military commission, they wrote.

At the time, they warned Iraqi courts might be unable to convict him, if Iraq gained custody. Daqduq, they said, might take advantage of ineffective incarceration or other Iraqi challenges.

If he is released from United States custody, there is little doubt that Daqduq will return to the battlefield and resume his terrorist activities against the United States and our interests, the lawmakers wrote.

(Editing by Warren Strobel and Cynthia Osterman)