Iraqi forces have entered the centre of Falluja, the Iraqi city longest held by Islamic State, nearly four weeks after the start of a US-backed offensive that cleared out the tens of thousands of residents still there.

Government troops, supported by multiple air strikes from a US-led coalition, recaptured the municipal building on Friday, though the ultra-hardline militants still controlled a significant portion of Falluja, an hour's drive west of Baghdad, and many streets and houses remain mined with explosives.

Federal police raised the Iraqi state flag above the government building and continued pursuing insurgents, according to a military statement. Hours later, a coalition spokesman said the advance was confirmed.

Prime Minister Haider al-Abadi said on Twitter that he would soon declare victory in Falluja, as government forces continued pushing into the city.

Troops could be seen coming under sniper fire earlier in the day as they entered a large mosque about 100 metres from the municipal building. Clashes also involved gun fire, artillery and aerial bombardment.

Heavily armed Interior Ministry police units were advancing along Baghdad Street, the main east-west road running through the city, and commandos from the counter-terrorism service (CTS) had surrounded Falluja hospital, the military statement said.

Sabah al-Numani, a CTS spokesman, said on state television that snipers were holed up inside the hospital but the facility was expected to be retaken within hours.

Iraq launched a major operation on May 23 to retake Falluja, a bastion of the Sunni Muslim insurgency against US forces that toppled Saddam Hussein in 2003 and Shi'ite-led governments that followed.

The participation of Iranian-backed Shi'ite militias in the battle alongside the Iraqi army raised fears of sectarian killings, and authorities are already investigating allegations that militiamen executed dozens of Sunni men fleeing the city.

Iraq's top Shi'ite cleric, Grand Ayatollah Ali al-Sistani, urged pro-government fighters in a Friday sermon not to seek revenge against residents.

Falluja was seen as a launchpad for recent Islamic State (IS) bombings in Baghdad, making the offensive a crucial part of the government's campaign to improve security in the capital.

Islamic State has begun allowing thousands of civilians trapped in central Falluja to escape and the sudden exodus has overwhelmed displacement camps already filled beyond capacity.

More than 6000 families left on Thursday alone, according to Falluja Mayor Issa al-Issawi, who fled following the IS seizure of city in January 2014.

"We don't know how to deal with this large number of civilians," he told Reuters on Friday.

The number of displaced people surpassed 68,000, according to the United Nations, which recently estimated Falluja's population at 90,000, only about a third of the total in 2010.