"The security forces are advancing towards central Fallujah from the southern side but doing so cautiously, to preserve civilian lives," Lieutenant-General Abdelwahab al-Saadi told AFP.
"In the coming days, we will declare the liberation of Fallujah," said Saadi, overall commander of the operation launched on May 22-23 to retake the jihadist bastion west of Baghdad.
The Joint Operations Command admitted that progress was being slowed by the astounding number of improvised explosive devices laid by IS in the city.
"There are tunnels and between 150 and 200 bombs are defused every 100 metres (yards)," it said in a statement.
"Our information is that IS has prepared car bombs, which they are hiding in homes with the intention of attacking us when we enter," the operations command said.
The Hashed al-Shaabi paramilitary force, which is dominated by Shiite militias that have so far operated on the periphery of the city, warned they would move in if the operation dragged on.
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A spokesman for the group put the number of Hashed al-Shaabi fighters involved at the beginning of the Fallujah operation at 30,000.
Some militias have been accused of sectarian violence against Sunni civilians during previous operations, and their involvement in an offensive against one of IS's most emblematic strongholds in the Sunni province of Anbar is seen as potentially explosive.
Elite forces have struggled to push deep into Fallujah over the past week, citing both tough resistance from IS and concern for the plight of an estimated 50,000 trapped civilians.
Close to 20,000 people have fled outlying areas but very few have been able to slip out of the centre, where IS is using them as human shields.
Residents trying to reach the safety of displacement camps set up south of Fallujah were taking massive risks to cross the Euphrates River.
"I saw three children being put in an open refrigerator so they could cross the river, but it sank and one of the children, a little girl, couldn't be saved so she died," a 45-year-old survivor told the Norwegian Refugee Council.
The International Rescue Committee said at least four people, three of them children, have drowned in the Euphrates trying to flee the conflict over the past few days.
In the holy Shiite city of Karbala, 100 kilometres (60 miles) to the south of Fallujah, a jihadist-claimed car bombing killed at least three people on Tuesday in the first such attack in more than two years.