US Central Command meanwhile said that in the past 24 hours US forces in al-Anbar province, where Fallujah is located, killed three Iraqis and captured 16 while suffering one US death. It did not give details about the American fatality.
A fourth “enemy personnel” was killed when US troops returned fire after an ambush, Centcom said.
The pair gunned down in Fallujah on Monday were shot inside a car, said Iraqi police Lieutenant Hamid Ali Bardi.
An American military spokeswoman in Baghdad said she was checking on the report but had no confirmation.
The shooting occurred about 1800 GMT, just after an explosion in the town, a centre of anti-coalition resistance 60 km west of Baghdad.
Central Command only said two “enemy” personnel died in an exchange of fire with a patrol of US paratroops.
US troops had earlier taken control of the sacked regional government offices and stationed themselves around the local Iraqi police station which had been deserted in the afternoon.
Military aircraft dropped flares and US soldiers used loudhailers to call on residents to give up their arms. They threatened to “shoot anyone seen with a gun in his hand.”
The US troops deployed after Saddam supporters stormed the government building, broke up the furniture, computers and air conditioning and then proceeded to destroy documents.
They then set all the wreckage alight in a huge bonfire outside, the correspondents said.
Supporters of the ex-dictator
Two large pictures of Saddam and Iraqi flags were hung from the top of the building.
No US soldiers were around the government offices in the Sunni-dominated town when the demonstration took place. The building is home to a committee responsible for reconstruction, overseen by American civilians.
“We can confirm that there was a pro-Saddam demonstration in Fallujah today,” the military spokeswoman said.
Fallujah, much of which remains loyal to Saddam, broke out in joyous displays earlier Monday after rumours spread that the man US forces captured Saturday night was not the former Iraqi leader.
Hundreds of local residents took to the streets, chanting pro-Saddam slogans and letting off celebratory gunfire into the air.
Several hundred armed demonstrators pledged their allegiance to their deposed leader, chanting, “We will sacrifice ourselves to you Saddam, with our blood, with our soul.”
Saddam surrendered late Saturday to US troops, who found him hiding in a hole outside the town of Ad Dawr, not far from his hometown of Tikrit in northern Iraq.
Near Ramadi, another centre of the insurgency west of Fallujah, a US patrol was ambushed with rocket-propelled grenades and small arms fire.
One insurgent died when the US unit returned fire, Central Command said.