The Iraqi army has stormed to the southern edge of Falluja under US air support.
It launched a direct assault to retake the city from Islamic State (IS) militants and help protect the nearby capital Baghdad from suicide bombings.
As government forces pressed their onslaught, a car bomb as well as suicide bombers driving a car and a motorcycle killed more than 20 people and injured over 50 in three districts of Baghdad, police and medical sources said.
Bolstered by Iranian-backed Shi'ite Muslim militia, the Iraqi army launched its operation to recover Falluja on May 23, first by tightening a six-month-old siege around the city 50km west of Baghdad.
Falluja in January 2014 became the first Iraqi city to fall to the ultra-radical Sunni Muslim IS, and it subsequently overran wide areas of the north and west of Iraq, declaring a caliphate that included seized territory in neighbouring Syria.
On Monday, army units advanced to the southern entrance to Falluja, "steadily advancing" under air cover from the US-led coalition, according to a military statement read out on state TV. A Reuters TV crew on the scene said explosions and gunfire were ripping through Falluja's southern Naimiya district.
A Shi'ite militia coalition known as Popular Mobilisation, or Hashid Shaabi, were seeking to consolidate the siege by dislodging militants from Saqlawiya, a village just to the north of Falluja.
The militias have pledged not to take part in the assault on the mainly Sunni Muslim city itself to avoid aggravating sectarian strife.
Falluja is a bastion of the Sunni insurgency that fought the US occupation of Iraq and the Shi'ite-led Baghdad government that took over after the fall of Iraqi strongman Saddam Hussein, a Sunni, in 2003.
The 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 far north that had a pre-war population of about 2 million.
Kurdish Peshmerga forces on Sunday launched an attack to oust Islamist militants from a handful of villages about 20km east of Mosul so as to increase the pressure on Islamic State and pave the way for storming the city.
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