BAGHDAD -- Iraq's government forces on Tuesday dislodged the Islamic State group from two northern neighbourhoods of Fallujah as an Iraqi military commander claimed the month-long offensive to recapture the city had left 2,500 IS militants dead.

The announcements came just days after the government had declared the liberation of Fallujah, the last bastion of the Islamic State group in the sprawling western Anbar province.

With aerial support from the U.S.-led coalition, Iraqi special forces took control of the neighbourhoods of al-Shurta and al-Jughaifi, special forces' Brig. Gen. Haider al-Obeidi told The Associated Press.

He said Iraqi military engineers were clearing the streets and buildings of left-over bombs.

Teaming up with paramilitary troops and backed by the U.S.-led coalition, Iraqi government forces launched the large-scale Fallujah operation in late May. On Friday, Prime Minister Haider al-Abadi declared victory after special forces entered the city centre, capturing government buildings and the central hospital.

Then, Iraqi commanders said 80 per cent of the city was under their control, though clashes were still underway in its northern parts.

In an interview with the local al-Sumaria TV late on Monday, the counterterrorism forces' chief in the Fallujah operation, Lt. General Abdul-Wahab al-Saadi, said about of 2,500 IS fighters have been killed in the offensive.

He offered no evidence to back up his claim and also said the number of IS fighters inside Fallujah had ranged between 3,500 to 4,000 when the offensive began. He claimed about 15 per cent of them were foreign fighters.

He cited Iraqi police reports as saying 1,086 IS-linked suspects have been arrested. He didn't say how many IS militants remain in Fallujah. Iraqi troops have not disclosed their losses in Fallujah, though the Islamic State group claims to have killed dozens

The operation has fueled an exodus of thousands of families, overwhelming camps for the displaced run by the government and aid groups.

In a briefing on Tuesday in Geneva, the UN refugee agency said more than 85,000 people have fled Fallujah and the surrounding area since the offensive began. UNHCR called for $17.5 million to meet the immediate needs of the growing number of displaced.

UNHCR spokeswoman Ariane Rummery said that she expected that thousands more "could still be planning to leave the city."

"These escalating needs have pushed UNHCR funding into crisis levels," Rummery said. "We are exhausting available resources in Iraq to deal with the rapid developments" in Fallujah.

The extremist group still controls Iraq's second-largest of Mosul and large parts of neighbouring Syria. According to UN figures, the violence has forced more than 3.4 million Iraqis to flee their homes. More than 40 per cent of the displaced are from Anbar province.

Associated Press writers Sinan Salaheddin in Baghdad and Jamey Keaten in Geneva, Switzerland contributed to this report