Although the offensive was launched on Monday, Iraqi government forces have besieged the city and its suburbs for several months now, resulting in shortages of food and medicines.
At least 15 Iraqi have died so far in the assault on the city in Anbar province.
On Tuesday local residents reported sporadic shelling around the city center, but said it was less intense than on the previous day.
The Iraqi said that overnight it had dislodged ISIL fighters from al-Karma, a village east of the city.
No casualties were reported by the or Fallujah’s main hospital.
The UN has said it is concerned about the fate of about 50,000 civilians still in the city as Iraqi forces battle fighters of the Islamic State of Iraq and the Levant (ISIL, also known as ISIS) group.
The Iraqi government says it is opening humanitarian corridors for civilians to flee the offensive – to the west, southwest and southeast of Fallujah.
From there, locals are being sent to makeshift camps.
But those being evacuated were asked to travel along this main road, riddled with booby-trapped improvised explosives and full of ISIL fighters.
Fallujah became in January 2014 the first Iraqi city to be captured by ISIL, six months before the group declared its caliphate.