Overnight, Iraqi soldiers gained control of the main railway station in Falluja.
In the run-up to the assault, troops also took two bridges and the city's main hospital.
They are now said to be advancing street-by-street towards the city centre.
Curfew
But there are concerns about thousands of regular people who remain in the city.
The Iraqi prime minister has imposed a curfew on the city and closed airports and borders as part of the assault.