But it's another group of Iraqis — using Syria as a base — who would prove key in ending the violence. Just days ago, more than 100 senior Iraqi Ba'athists met there just days ago with the backing of Syrian authorities.
They are former members of Saddam's regime, many involved in organizing and financing the insurgency, whose attacks on both Iraqis and Americans have killed thousands.
But the Ba'athists now claim they are willing to trade this violence for political discussions, after secret meetings in Syria.
In an exclusive interview with CBS News, Syria's Vice President Farouk al Shara says negotiations are the way forward.
"If they are flexible and if they really believe in the political process, then why not," he says.
The Syrians are not only trying to nudge the old Ba'athists into negotiations, but the Shiite militias too, the most dangerous of which is controlled by Moqtada al Sadr, who was in Damascus last month and is due to come back soon.
Extended Video: The al Shara interview
"We don't want to exaggerate our role. But it is important, it is significant," al Shara says.
So where does that leave Washington?
"My attitude on Syria is they can be a much more constructive partner and they haven't been," President Bush has said.
But so far, they're not interested.
"Coming to Syria is not a concession. Coming to Syria is an added value," al Shara says.
While the American discussion about the Iraq war focuses on how many troops to put in or take out right now, there appears be a new strategy to consider – whether American diplomats should go to Syria to join talks that could lead to a possible political progress.