U.S. Defense Secretary Robert Gates (L) talks with Saudi Assistant Minister of Defense and Aviation Prince Khalid bin Sultan during a coffee ceremony after Gates's arrival in Riyadh. (Chip Somodevilla/GETTY IMAGES)

Defense Secretary Robert M. Gates met with King Abdullah bin Abdul Aziz and other top Saudi officials Wednesday, at the start of his third trip to the tumultuous region in the past month.

“We had a very good meeting,” Gates told reporters traveling with him, after speaking with the king for 90 minutes. “It was an extremely cordial, warm meeting. I think the relationship is in a good place.”

In addition to Abdullah, Gates met with Prince Khalid bin Sultan, one of the Saudi government’s top defense officials.

Late Wednesday, he traveled to Baghdad for what is widely expected to be his final trip to Iraq as defense secretary. During his three-day trip, he is to meet with Prime Minister Nouri al-Maliki and U.S. troops and visit northern Iraq.

On the agenda for Gates’s meetings in Baghdad is the question of what role the U.S. military will play after the end of the year, when the planned troop withdrawal is scheduled to be completed. A senior defense official traveling with Gates said that if Iraq wants some small U.S. force to remain, the request should come soon.

“The ball is in the Iraqis’ court,” the official said, speaking on the condition of anonymity. Gates “has made it pretty clear that these things don’t happen overnight.”

U.S. relations with the Saudis have been strained in recent months amid spreading unrest across the Arab world. Saudi officials have voiced unhappiness that the Obama administration urged the departure of Egyptian President Hosni Mubarak, a longtime ally, following a popular uprising.

And in March, the Saudis sent troops and tanks into neighboring Bahrain to help quell demonstrations at a time when U.S. officials were advocating an approach based on negotiations. Bahrain is home to the U.S. Navy’s 5th Fleet and is regarded as a strategically important bulwark against Iranian power in the region.

During Wednesday’s visit, however, U.S. defense officials minimized reports of tensions with the Saudis. Gates said he did not raise any concerns with Abdullah about the Saudi troops in Bahrain.

Gates had also planned to report to Abdullah on the progress of a $60 billion arms deal with the Saudis and to discuss plans for upgrading the nation’s missile defense system, U.S. officials said. Saudi Arabia is the largest buyer of U.S. arms. Iran, which officials say is seeking to exploit the political upheaval in the Arab world, was on the agenda for discussions, as well.

At a time when U.S. officials have sharply criticized other Arab nations for their handling of pro-democracy unrest in the region, Gates planned to urge “evolutionary” reform and respect for human rights in Saudi Arabia. There have been only small protests here so far.

The senior defense official said the United States needed to take a “pragmatic approach” in dealing with such a close ally.

“They’re going to have to find their own path,” the official said.

Gates made two visits to the region last month, stopping in Egypt, Israel, Jordan and Bahrain. He left Bahrain expressing optimism that the government there would take a cautious approach to pro-democracy demonstrations, but the Saudi troops entered the country days later, and the violent crackdown followed soon afterward.

Gates was also supposed to visit Saudi Arabia last month, but that visit was canceled because of Abdullah’s ill health, U.S. officials said. The king’s condition has since improved, the officials said.