The Iraqi army stormed to the southern edge of Falluja under US air support today and captured a police station inside the city limits, launching a direct assault to retake one of the main strongholds of Islamic State militants.

A Reuters TV crew about a mile (about 1.5 km) from the city's edge said explosions and gunfire were ripping through Naimiya, a district of Falluja on its southern outskirts.

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

The battle for Falluja is shaping up to be one of the biggest ever fought against Islamic State, in the city where US forces waged the heaviest battles of their 2003-2011 occupation against the Sunni Muslim militant group's precursors.

Falluja 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 Shi'ite civilians and government targets inside the capital.

As government forces pressed their onslaught, suicide bombers driving a car and a motorcycle and another bomb planted in a car 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.

The Iraqi army launched its operation to recover Falluja a week ago, first by tightening a six-month-old siege around the city 50 km (30 miles) west of Baghdad.

Falluja, in the heartland of Sunni Muslim tribes who resent the Shi'ite-led government in Baghdad, was the first Iraqi city to fall to Islamic State in January 2014. Months later, the group overran wide areas of the north and west of Iraq, declaring a caliphate including parts of neighbouring Syria.

On Monday, army units advanced to the city's southern entrance, "steadily advancing" under air cover from a US-led coalition helping to fight against the militants, according to a military statement read out on state TV.

A Shi'ite militia coalition known as Popular Mobilisation, or Hashid Shaabi, was seeking to consolidate the siege by dislodging militants from Saqlawiya, a village just to the north of Falluja.

The militias, who took the lead in assaults against Islamic State in other parts of Iraq last year, have pledged not to take part in the assault on the mainly Sunni Muslim city itself to avoid aggravating sectarian strife.

Islamist militant stronghold

Falluja has been a bastion of the Sunni insurgency that fought both the US occupation of Iraq and the Shi'ite-led Baghdad government that took over after the fall of dictator Saddam Hussein, a Sunni, in 2003.

American troops suffered some of their worst losses of the war there in two battles in 2004 to wrest it back from al Qaeda in Iraq, the insurgent group now known as Islamic State.

The latest offensive is causing alarm among international aid organisations over the humanitarian situation in the city, where more than 50,000 civilians remain trapped with limited access to water, food and health care.

Falluja is the second-largest Iraqi city still under control of the militants, after Mosul, their de facto capital in the north that had a pre-war population of about 2 million.

It would be the third major city in Iraq recaptured by the government after Saddam's home town Tikrit and Ramadi, capital of Iraq's vast western Anbar province.

Falluja is also in Anbar, located between Ramadi and Baghdad, and capturing it would give the government control of the major population centres of the Euphrates River valley west of the capital for the first time in more than two years.

On the northern front, the security forces of the autonomous Kurdish region launched an attack on Sunday to oust Islamist militants from villages about 20 km (13 miles) east of Mosul so as to increase the pressure on Islamic State and pave the way for storming that city.

The Kurdish forces, known as peshmerga, have retaken six villages in total since attacking Islamic State positions on Sunday with the support of the US-led coalition, the Kurdistan Region Security Council said on Monday. That represents most of the targets of their latest advance.

Prime Minister Haider al-Abadi hopes to recapture Mosul later this year to deal a decisive defeat to Islamic State.

Abadi announced the onslaught on Falluja on May 22 after a spate of bombings that killed more than 150 people in one week in Baghdad, the worst death toll so far this year. The worsening security in the capital has added to political pressure on Abadi, struggling to maintain the support of a Shi'ite coalition amid popular protests against an entrenched political class.

Monday's bombings targeted two densely populated Shi'ite districts, Shaab and Sadr City, and a government building in one predominantly Sunni suburb, Tarmiya, north of Baghdad.

A car bomb in Shaab killed 12 people and injured more than 20, while in Tarmiya eight were killed and 21 injured by a suicide bomber who pulled up in a car outside a government building guarded by police. In Sadr City, a suicide bomber on a motorcycle killed three people and injured nine.