A military unit is trying to advance in the city, he said as explosions and gunfire could be heard in Falluja's southern Naimiya district.
The Iraqi army, supported by Iranian-backed Shi'ite militia, began the operation to recapture Falluja 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, Falluja 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.