BAGHDAD (AFP) -- Iraqi forces in Fallujah are facing resistance from holdout Islamic State group fighters in just two northern neighbourhoods of the city, the operation's overall commander said on Wednesday.

"The northern and central parts of Fallujah have almost been cleared of Daesh," Lieutenant General Abdulwahab al-Saadi told AFP, using an Arabic acronym for IS.

"There are few IS fighters left, only in the Al-Muallemin and Jolan neighbourhoods in the north of the city," he said.

"The militants in Jolan are offering some resistance but we're pushing back and we've killed a number of them," he said.

Operations against IS in northern Fallujah were being conducted by the elite counter-terrorism service and forces from the federal and provincial police.

Prime Minister Haider al-Abadi launched the offensive against the jihadist stronghold, 50 kilometres west of Baghdad, a month ago.

After an initial phase of staging operations to encircle Fallujah, elite federal forces stormed the city centre and were able to gain the upper hand relatively quickly.

Abadi declared victory on June 17, saying only small pockets of IS fighters remained after Iraqi forces raised the national flag over the main government compound in the city centre.

Saadi and other Iraqi commanders have said government forces controlled at least three-quarters of the city.

Christopher Garver, the spokesman of the US-led coalition assisting Iraqi forces, said Tuesday that by the US military's definition, only a third of the city had been cleared.

US forces battling one of IS's previous incarnations in Fallujah in 2004 suffered some of their worst losses since the Vietnam War, despite huge numerical and technological superiority.

Iraqi forces that have been reconquering swathes of territory lost to IS two years ago had been expected to face their toughest battle yet and IS fighters to defend their emblematic bastion to the death.

Yet after breaching the jihadists' defences in the south of the city, Iraqi forces have moved relatively rapidly and, despite persistent violence in northern neighbourhoods, the outcome of the battle appears in no doubt.