The Iraqi army has advanced to the southern edge of Fallujah under US air support and captured a police station inside the city limits, launching a direct assault to retake one of the main strongholds of the so-called Islamic State.

Explosions and gunfire were heard through Naimiya, a largely rural district of Fallujah on its southern outskirts.

An elite military unit, the Rapid Response Team, seized the district's police station, state TV reported.

The unit advanced northward, stopping about 500m from the al-Shuhada district, the southeastern part of city's main built-up area, army officers said.

Fallujah is Islamic State's closest bastion to Baghdad, and believed to be the base from which the group has plotted an escalating campaign of suicide bombings against Shia civilians and government targets inside the capital.

As government forces pressed their onslaught, suicide bombers driving a car and a motorcycle blew themselves up in the capital.

Along with another bomb planted in a car, they killed more than 20 people and injured more than 50 in three districts of Baghdad, police and medical sources said.

Separately, Kurdish security forces announced advances against Islamic State in northern Iraq, capturing villages from militants outside Mosul, the biggest city under militant control.