The fighting prompted a new exodus of thousands of desperate civilians and deep concern for the many more trapped in the battlegrounds.
The overall commander of the Fallujah operation, Abdelwahab al-Saadi, said Saturday it was a matter of hours before the Counter-Terrorism Service (CTS) entered the city.
The week-old operation has so far focused on retaking villages and rural areas around Fallujah, which lies just 50 kilometres (30 miles) west of Baghdad.
CTS's involvement will mark the start of a phase of urban combat in a city where US forces in 2004 fought some of their toughest battles since the Vietnam War.
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The jihadists were also under pressure from Kurdish fighters east of their northern Iraqi stronghold Mosul and from US-backed Kurdish-led fighters in Syria.
Iraq's autonomous Kurdish region on Sunday announced the launch of a pre-dawn offensive involving 5,500 peshmerga fighters to retake an area on the road between its capital Arbil and Mosul.
Ten hours into the operation, which was launched a day after a wave of 12 coalition air strikes in the area, Kurdish forces had fully retaken three villages, it said.
In Syria, Kurdish rebels from the People's Protection Units (YPG) allied to Arab fighters and backed both on the ground and in the air by the US-led coalition, were targeting Raqa, IS's de-facto Syrian capital.