Forces from Iraq's 8th Division backed by tribal fighters set out from the Amriyat al-Fallujah area south of the city and the Al-Salam intersection to its southwest, said Staff Major General Ismail al-Mahalawi, the head of the Anbar Operations Command.
US-led coalition and Iraqi forces are providing air support, Mahalawi said.
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Prime Minister Haider al-Abadi announced the start of the long-awaited operation to retake the city early on Monday and less than a day into the battle, Iraqi forces had secured the nearby town of Garma.
That cut off IS fighters in Fallujah from one of their last support areas and paved the way for more advances towards the city, which lies only 50 kilometres (30 miles) west of Baghdad.
But with forces converging on the city, concerns mounted that the tens of thousands of civilians believed to still be inside had nowhere to go.
Fallujah was seized by anti-government fighters in early 2014 and later became an IS bastion. It is one of two major Iraqi cities still held by the jihadists.