An AFP photographer embedded with interior ministry SWAT teams said Iraqi forces had managed to enter the Shuhada neighbourhood, on the southern edge of the city centre.
"The security forces have advanced from Naimiya neighbourhood to Shuhada," Lieutenant General Abdelwahab al-Saadi, the operation's overall commander, told AFP.
"The clashes between the security forces and Daesh members were fierce," said another officer, using an Arab acronym for IS.
Prime Minister Haider al-Abadi announced the launch of an operation to retake Fallujah on May 23.
After an initial phase of shaping operations aimed at cutting off the jihadists' last supply lines, elite forces on May 30 attempted to enter Fallujah from three directions.
The Norwegian Refugee Council, which runs camps for displaced people south of the city, said today that an estimated 12,000 residents of the area have fled their homes since May 21.
None of them however were from the city centre, where trapped civilians have little to eat, no safe water to drink and are at risk from intense shelling and bombardment.
The city has been besieged for months and no aid delivery has reached it since September 2015.
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