The two countries are in negotiations over this and top US military commander Admiral Mike Mullen is in Baghdad to urge Iraqi leaders to spur up discussions if they want US forces to stay beyond the end of 2011, `Wall Street Journal` reported.
The US currently has 50,000 troops stationed in Iraq, down from a peak of nearly 170,000 following the US led invasion in 2003.
The paper quoting unnamed officials said US military commanders believe that leaving at least 10,000 troops beyond 2011 could promote greater security and prevent Iran from expanding its regional influence.
US allies Saudi Arabia and Israel have echoed the concern that if US pulls out completely, Iran could extend its influence.
The talks between US military commanders and Iraqi officials come as Washington is scheduled to start drawing down remaining forces in late summer or early fall.
While American defence officials have made it clear that they want to leave troops in Iraq, WSJ quoting officials said such a decision would require Presidential approval.
US Defence Secretary Robert Gates who recently made a surprise visit to Baghdad said that American forces were prepared to stay in any role beyond the schedule pull out, but time was running out for Baghdad to ask for it.
WSJ said Iraqi officials were concerned that a lingering US troops presence would fuel a popular revolt like that has been convulsing the region in recent months and which has swept aside powerful Presidents in Egypt and Tunisia.