BAGHDAD – Face to face with Iraq’s leaders, Barack Obama gained fresh support Monday for the idea of pulling all U.S. combat forces out of the war zone by 2010. But the Iraqis stopped short of actual timetables or endorsement of Obama’s pledge to withdraw troops within 16 months if he wins the presidency.

The Democratic presidential contender also got a military briefing – and a helicopter tour – from the top U.S. commander in the region, Gen. David Petraeus, and also met with a few of the nearly 150,000 U.S. troops now well into the war’s sixth year.

Back in the U.S., Republican rival John McCain said he hoped Obama’s visit would open his eyes to the danger of withdrawal timetables. Said the Arizona senator, who was meeting with President Bush’s father, the former president, in Maine: “When you win wars, troops come home.” He said of Obama: “He’s been completely wrong on the issue.”

In Washington, the White House expressed displeasure with recent public comments by Iraqi leaders on the withdrawal question and suggested they might have the U.S. election on their minds.

As Obama visited Iraq for the first time in more than two years, comments Monday by the government’s spokesman roughly mirrored the Illinois senator’s withdrawal schedule and offered a glimpse of Iraq’s growing confidence as violence drops and Iraqi security forces expand their roles.

“We are hoping that in 2010 that combat troops will withdraw from Iraq,” spokesman Ali al-Dabbagh said after Obama met with Prime Minister Nouri al-Maliki – who has struggled for days to clarify Iraq’s position on a possible timetable for a U.S. troop pullout.

Iraq’s Sunni vice president, Tariq al-Hashemi, said after meeting Obama that Iraqi leaders share “a common interest … to schedule the withdrawal of American troops.”

“I’d be happy if we reach an agreement to say, for instance, the 31st of December 2010” would mark the departure of the last U.S. combat unit, he said – then noted that any such goal could be revised depending on threats and the pace of training for Iraqi security forces. That date would be some seven months later than Obama’s 16-month timeline.

Obama said almost nothing to reporters as he walked to and from his meetings, but promised fuller impressions after he finishes here Tuesday and he heads to Jordan and Israel.

But he told ABC News that military leaders have “deep concerns” about a timetable that doesn’t account for changing conditions.

“I don’t think that there are deep concerns about the notion of a pullout per se,” he said in the interview. “There are deep concerns from their perspective about a timetable that doesn’t take into account what they anticipate might be some sort of change in conditions.”

Obama also told ABC that knowing what he knows now he still would have opposed sending more troops to Iraq last year.