US military presence could be open-ended under future deal between Bush and Maliki.
About 160,000 US troops are in Iraq under a United Nations mandate enacted after the US-led invasion in 2003.
The US has repeatedly denied seeking permanent bases in Iraq or that the US deployment was open-ended.
But in November leaders of the two countries signed a deal setting the ground rules for friendly long-term ties.
Under the “declaration of principles” the two sides will determine how many US troops will remain in Iraq and the legal framework that will govern their presence in the country.
The US says Iraq has become less violent in recent months after additional troop deployment.
Washington plans to withdraw more than 20,000 troops by June next year.