BAGHDAD (AP) — The Latest on Iraq's battle against the Islamic State group (all times local):

5:15 p.m.

The U.S.-led coalition says Fallujah is approximately one-third "cleared" of the Islamic State group, days after the Iraqi government declared victory in the city west of Baghdad.

U.S. Army Col. Christopher Garver, the Baghdad-based spokesman for the coalition, says other parts of the city are "contested," with clashes underway between Iraqi security forces and IS fighters.

Garver says most of the cleared terrain is in the south of the city and "clearing operations continue outward from the city center."

Iraqi commanders on the ground have said 80 percent of the city is under their control.

Fallujah is the last bastion of the Islamic State group in the sprawling western Anbar province. IS still controls pockets of territory in Iraq's north and west, including Mosul, the country's second largest city.

1 p.m.

A senior Iraqi military commander says about 2,500 Islamic State militants have been killed during a monthlong offensive to recapture the city of Fallujah.

The counterterrorism forces' chief in the operation, Lt. General Abdul-Wahab al-Saadi, told the local al-Sumaria TV late on Monday that the number of IS fighters inside Fallujah ranged between 3,500 to 4,000 when the offensive began in late May.

Iraqi troops have not disclosed their losses in Fallujah though the Islamic State group claims to have killed dozens

Al-Saadi offered no specifics to back up the figure of 2,500 killed IS fighters. He claimed about 15 percent 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.