Recalling the enormous American investment in Iraq, Defense Secretary Robert Gates said Tuesday he hopes Baghdad asks U.S. troops to stay beyond their scheduled Dec. 31 departure in order to preserve the relative peace.

"I hope they figure out a way to ask, and I think that the United States will be willing to say yes when that time comes," Gates said in response to a question about Iraq after delivering a speech on Pentagon budget cuts.

Gates said a longer U.S. military presence could help sustain the security and other gains Iraq has made in recent years. Iraq could become a model for a multisectarian society in the Arab world "that shows that democracy works," he said.

The Pentagon chief's comments on Iraq were in line with what he said while visiting the country last month. During that trip, he said Iraqi political and military leaders acknowledge they need further U.S. military support but are reluctant to ask for it because popular sentiment is strongly against an extended U.S. presence.

In his remarks Tuesday he was more expansive, asserting that a longer U.S. stay would send a reassuring signal to other Persian Gulf states facing civil unrest and heightened concern about the influence of Iran. Gates also has acknowledged that Iraqi domestic politics make it difficult to seek continued military assistance.

"Whether we like it or not, we're not very popular there," Gates said at the American Enterprise Institute, a think tank. A member of his audience had asked whether he thought it was in America's interest to say in Iraq longer.

"We've made a big investment already, a huge investment in treasure and in lives," he said. In the more than eight years since U.S. forces invaded to topple Saddam Hussein, the U.S. has lost more than 4,400 lives, with more than 32,000 wounded.

Gates was bullish on Iraq's political prospects.

"Something we could not have predicted five months ago is that Iraq would emerge as the most advanced Arab democracy in the entire region," he said.

Violence has decreased dramatically in Iraq since the height of the war, but bombings and shootings still happen daily across the country. On Sunday, 16 people were killed in a series of blasts in and around the capital. Many Iraqis worry that violence may increase when U.S. troops pull out.

There currently are about 47,000 U.S. troops in Iraq, none in a declared combat role. All are scheduled to leave by Dec. 31 under an agreement worked out in 2008 shortly before President Barack Obama entered the White House. Obama has repeatedly said he intends to carry out the current agreement.

Behind the scenes, Pentagon officials are trying to sort out how they would manage an extended troop presence, given that current plans call for a rapid drawdown of forces and equipment in early fall. Adm. Mike Mullen, chairman of the Joint Chiefs of Staff, said in Baghdad on April 22 that Iraq must make up its mind in a matter of weeks.

Gates said the U.S. could provide further assistance to the Iraqi military, including helping secure their air space, "at relatively small cost to ourselves _ especially given the investment that's already been made."

Separately, the Pentagon on Tuesday announced that the headquarters of the Army's 3rd Infantry Division _ the unit that spearheaded the U.S. drive into Baghdad in March 2003 _ will return to Iraq this summer as part of a regular rotation. The headquarters, based at Fort Stewart, Ga., will deploy with 775 personnel, along with two Army brigades totaling 6,400 soldiers.

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Robert Burns can be reached at http://twitter.com/robertburnsAP