Iraqi special forces, also known as its counterterrorism forces, have secured the largely agricultural southern neighbourhood 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 neighbouring 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.
At the same time, Syrian government troops are advancing on Raqqa from the south.
A top Syrian Kurdish commander died last night, several days after sustaining injuries during a US-backed campaign to unseat the Islamic State group (IS) from its Raqqa stronghold.
Abu Layla, who commanded a brigade inside the predominantly-Kurdish Syrian Democratic Forces (SDF), was hit by IS sniper fire on the outskirts of Manbij, an IS-controlled areawhich is key to the supply route between the Turkish border and Raqqa.
He was evacuated by US forces to a hospital in the Iraqi Kurdish city of Sulaymaniyah, where he died.
Nearly 50 air strikes hit rebel-held areas in and around the Syrian city of Aleppo on Sunday in some of the heaviest recent raids by Russian and Syrian government aircraft, residents and a monitoring group said.
The Syrian Observatory for Human Rights also said an unidentified war plane had crashed in countryside south of Aleppo, in an area where Islamist rebel fighters are battling the Syrian army and Iranian-backed forces.
It had no information on what caused the crash.
A civil defence worker said at least 32 people were killed in the rebel-held parts of the city during the air strikes, with 18 bodies pulled from flattened buildings in the Qatrji neighbourhood.
The slow-moving Iraqi operation was announced in May. An array of troops had cleared IS from the majority of Fallujah’s suburbs.
Last Monday, Iraq’s special forces began pushing into the city centre, but they have faced stiff resistance as Fallujah has been under IS control for more than two years, and the militants have erected complex defences.
“Vibed! Vibed!” shouted an Iraqi air commander from a small mobile base on Fallujah’s southern edge.
Using an acronym for a car bomb, the Iraqi special forces officer called to Australian coalition forces over a hand-held radio.
Moments later, plumes of 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.
Al-Saadi says coalition air power in Fallujah has prevented car bombs from inflicting casualties, but they have still succeeded in slowing progress.
“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.