U.S. President George W. Bush opposes a specific timetable for pulling out the troops, insisting such a move hinge on situation evolvement on the ground.
In a video conference last week, Bush and Maliki agreed on a "time horizon" for reducing the troops.
Bush sent in five combat brigades last year to quell a growing wave of violence in Iraq. Now, violence here has dropped to a four-year low.
The last batch of reinforced American troops is expected to leave by the end of this month. U.S. military commanders are mulling on further cut of force according to an assessment of the local security situation.
General David Petraeus, U.S. top commander in Iraq, is expected to make his recommendations on future troop levels in a report to the U.S. Congress in September.
Also, Adm. Mike Mullen, chairman of the Joint Chiefs of Staff, has mentioned a perspective of further drawdown following his recent visit to Iraq.
While noting the security gains are not irreversible, Mullen said on Wednesday that the situation is "unquestionably and remarkably better," and "if these trends continue I expect to be able to recommend to the secretary and the president further troop reductions early this fall."
During his stay in Afghanistan, Obama met with President Hamid Karzai and visited U.S. military bases.
Before coming to Baghdad, Obama visited the southern Iraq city of Basra. The oil-rich region has long been a hotbed of turf war among Shiites, and saw large-scale crackdown operations against militants in March.
In Iraq, the locals were divided over their preference for the next U.S. president.
"I am not really care who will be the next U.S. president, because I think the policy of the U.S. administration would not be affected by a person," said Dhiyaa al-Hadithy, a 38-year-old physician, "However, if you insist, I prefer Obama, because this man supports the U.S. troop withdrawal from Iraq as soon as possible. Besides, the man is the son of a black Kenyan. Maybe he will feel our suffering because of his cultural background."
