Iraqi police say they want to keep a safe distance from US troops for fear of getting caught in the crossfire.
US occupation forces have been repeatedly targeted by snipers and guerillas using rocket-propelled grenades to attack their convoys.
Dozens have been killed and wounded since George Bush declared major combat officially over on May 1st.
The town of Fallujah, west of Baghdad, has been a centre of much of the resistance to the coalition forces. It has seen several deadly attacks on US and Iraqi forces since US troops killed 20 protesters there in April.
US soldiers have been using a police station in the town as a base, but the new Iraqi policemen - most wearing uniforms supplied by the United States - say they will quit unless the Americans find a new base in 48 hours.
But the commander of US ground forces in Iraq insisted his men would not leave the police station. He said that if the Iraqis follow through with their threat, "we will find some more" policemen to patrol Fallujah.
Elsewhere, several Iraqi civilians were caught in the middle of a battle after assailants fired a rocket-propelled grenade at a US military patrol on the road to Baghdad International Airport overnight.
One Iraqi was shot in the neck and the other in the abdomen; their condition was not clear. The airport road is a frequent site of attacks on US troops.
AP