Pentagon spokesman Col. Robert Manning said in a press briefing on Monday he would not speculate on reports claiming that the Kurdish Pershmerga were pushed out of a military base known as K-1 located on the outskirts of Kirkuk.
Monday's Peshmerga statement accused a group within the PUK of "treason" for assisting Baghdad's advance.
The exodus in buses and cars towards Arbil and Sulaimaniyah, the two main cities of autonomous Iraqi Kurdistan, created traffic jams on roads leading out of Kirkuk. He feared the referendum, guaranteed to alienate all the Kurds'allies, would turn out to a political error with similar calamitous consequences.
He did so by holding a referendum on Kurdish independence on 25 September that was greeted with enthusiasm by Iraqi Kurds. The Kurds seized Kirkuk city in 2003 at the time of the United States invasion and expanded their area of control in 2014 when the Iraqi army in northern was defeated by Isis.
Mr Abadi told his security forces in a statement read on state television "to impose security in Kirkuk in cooperation with the population of the city and the Peshmerga".
The Kurdistan Region Security Council said early Monday that the peshmerga destroyed at least five USA -supplied Humvees being used by Iraq's state-sanctioned militias.
Brigadier General Bahzad Ahmed said Iraqi troops had "burnt lots of houses and killed many people" in Tuz Khurmatu and Daquq in the city's south.
Baghdad insisted the city and its province be returned, but matters came to a head when the Kurdish authorities expanded their referendum to include Kirkuk.
Exactly a year after US -backed Iraqi and Kurdish forces launched an operation to rout the Islamic State group and retake Mosul, Iraq's second-largest city, those USA -trained forces were pitted against each other in northeast Iraq on Monday.
"We're leaving because we're scared there will be clashes" in the ethnically mixed city of 850,000 people, said 51-year-old Chounem Qader.
The US-led coalition against IS urged the two sides to "avoid escalatory actions" and to focus on fighting the extremists, who are on the verge of losing their last strongholds in Iraq.
A statement by the US-led global military task force in Iraq described the clashes outside Kirkuk as a "misunderstanding".
The "government of Abadi bears the main responsibility for triggering war on the Kurdistan people, and will be made to pay a heavy price", the Peshmerga command said in a statement, cited by Kurdish leader Barzani's assistant Hemin Hawrami.
The military action in Kirkuk helped spur a jump in world oil prices on Monday.
Kurdish peshmerga troops who are defending the area are also US -armed and trained.
Last Friday, US President Donald Trump announced that the IRGC was the target of new American sanctions, describing Iran's leading military institution as the "corrupt personal terror force and militia" of the regime's supreme leader that has "hijacked" large portions of the economy "to fund war and terror overseas". Washington, allied with the Kurds for decades, had pleaded in vain for them to cancel the vote, arguing that it could lead to regional war and the breakup of Iraq. Kirkuk province underwent "Arabization" under former Iraqi dictator Saddam Hussein's Ba'ath Party, shifting the demographic balance of the province that was once majority Kurdish.
Swiftly. Iraq's army, its anti-terrorism forces and the federal police began their operations before dawn Monday. An Iraqi Oil Ministry official said that it would be "a very short time" before the Iraqi military seized all the oilfields in Kirkuk province.
Residents of Kirkuk said there was no sign the Iraqi forces were getting close to the city itself, which is under the control of the Kurdish Asayish police.
Kurdish leaders we consider as our brothers have agreed to hand over control of North Oil and North Gas company facilities who belong to the state, ' said a military commander involved in the operation.
Both Kurdish parties control their own Peshmerga units.
The PUK had supported a UN-backed plan for negotiations with Baghdad in exchange for dropping the referendum.
"The war on ISIS is edging to an end and now the real war starts, the war between the regional powers in order to control resources and define their own areas of influence", said Sami Nader, head of the Levant Institute for Strategic Affairs in Beirut.
Although Iraqi officials portrayed the Kurds as retreating without a fight, Kurdish officials said Peshmerga clashed with the "Popular Mobilisation", Shiite forces trained and armed by Iran that operate alongside regular Iraqi troops.