Iraqi special forces, also known as its counterterrorism forces, have secured the largely agricultural southern neighborhood of Naymiyah under cover of US-led coalition airstrikes, Lt Gen Abdel Wahab al-Saadi said.
Special forces are now poised to enter the main city, al-Saadi said.
The Fallujah operation coincides with a twin offensive on IS-strongholds in neighboring Syria. Syrian Kurdish forces are advancing on Manbij, an IS-held city controlling the supply route between the Turkish border and the town of Raqqa, the militants' de facto capital.
The slow-moving Iraqi operation was announced in May. An array of troops including Iraqi military divisions, the federal police and the largely Shiite militia groups, known as the Popular Mobilization Forces, had cleared IS from the majority of Fallujah's suburbs.
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On Monday, Iraq's elite special forces began pushing into the city center, but they have faced stiff resistance as Fallujah has been under IS control for more than two years, and the militants have been able to erect complex defenses.
Tall dirt berms dot the dusty fields to the city's south. A single column of counterterrorism Humvees snaked up toward a row of low lying houses that mark the beginning of the main city.
Moments later, a plume of white, then black smoke appeared on the horizon. Commanders at the scene said the explosion was created by a coalition rocket destroying the incoming car bomb.
Car bombs were once the most deadly form of IS counterattack for Iraq's special forces, who have taken the lead in a number of anti-IS operations, including in the cities of Tikrit and Ramadi.
"We are expecting many more," once inside the city's more urban neighborhoods, al-Saadi said.
Fallujah is one of the last strongholds of IS in Iraq. While the militants once held nearly a third of the country's territory, their grip has slipped to less than half that, according to the Iraqi Prime Minister Haider al-Abadi.
IS still controls patches of territory in northern and western Iraq, as well as its second largest city, Mosul.