A military unit is trying to advance in the city, he said as explosions and gunfire could be heard in Fallujah's southern Naimiya district.
The Iraqi army, supported by volunteer forces, began the operation to recapture Fallujah on May 23, first by tightening its siege around the city, 50 km (30 miles) west of Baghdad, and has now begun a direct assault.
In January 2014, Fallujah became the first Iraqi city to fall under the control of the militants, six months before they declared a caliphate over territory seized in Iraq and Syria.