Soldiers from the elite Rapid Response Team stopped their advance overnight about 500 meters (yards) from the al-Shuhada district, the southeastern part of city's main built-up area, an army commander and a police officer said, Reuters reported.
"Our forces came under heavy fire, they are well dug in in trenches and tunnels," said the commander speaking in Camp Tariq, the rear army base south of Fallujah, 50 km (30 miles) west of Baghdad.
A staff member of Fallujah's main hospital said they received reports of 32 civilians killed on Monday. The toll could not be confirmed by a second source.
Medical sources in the city had reported that the death toll in the city stood at about 50, 30 civilians and 20 militants, during the first week the offensive which started on May 23.
Fallujah has been under siege for more than six months. Foreign aid organization are not present in the city, but are providing help to those who manage to exit and reach refugee camps.
The latest offensive is causing alarm among these organizations as more than 50,000 civilians remain trapped with limited access to water, food and health care.
Fallujah is the second-largest Iraqi city still under control of the militants, after Mosul, their de facto capital in the north that had a pre-war population of about 2 million.
Iraq Prime Minister Haider al-Abadi announced the assault on Fallujah on May 22 after a spate of bombings that killed more than 150 people in one week in Baghdad, the worst death toll so far this year. A series of bombings claimed by Daesh also hit Baghdad on Monday, killing over 20 people.
Fallujah was the first city to fall under Daesh control, in January 2014.