Monday's Peshmerga statement accused a group within the PUK of "treason" for assisting Baghdad's advance.
Al-Assadi said the state-backed militias, mainly Shiite Arab fighters known as the Popular Mobilisation Forces, have not entered the city centre, but reporters saw Turkmen PMF militiamen taking up posts in the western part of the city.
McCain called for Kurdish and Iraqi leaders to "engage in a dialogue about the Kurdish people's desire for greater autonomy from Baghdad at an appropriate time and the need to halt hostilities immediately".
Kirkuk has found itself at the heart of a long-running dispute between Iraq's autonomous Kurdish region and its central government that reached fever pitch after Kurdish authorities staged a non-binding independence vote in late September.
An Iraqi army convoy lines up south of Kirkuk city.
Peshmerga forces loyal to the Patriotic Union of Kurdistan (PUK), a political party linked to Iraqi President Fuad Masum, who is himself a Kurd, were reported to be withdrawing from areas under their control after the operation was launched.
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The Iraqi National Security Council calls this a "declaration of war towards Iraqis and government forces".
The Iraqi parliament asked Mr Abadi to deploy troops to Kirkuk and other disputed areas after the referendum result was announced, but he said last week that he would accept them being governed by a "joint administration" and that he did not want an armed confrontation.
Thousands of residents were seen fleeing Kurdish districts of Kirkuk City, heading in buses and cars toward the autonomous Kurdistan region.
Denouncing the "plots by the Iraqi government and the Islamic Revolutionary Guards", the Peshmerga statement added that the Iraqi government should pay a "heavy price" for the attack on Kirkuk.
"We call on all parties to immediately cease military action and restore calm while we continue to work with officials from the central and regional governments to reduce tensions and avoid and futher clashes", the United States embassy said. The U.S. has armed, trained and provided vital air support to both sides in their shared struggle and called the frictions a distraction against the most important fight.
The short suspension in production helped push up world oil prices as the shutdown represented more than half of total Kurdish output. But it was adamantly opposed by the Iraqi central government, Iran, Turkey as well as traditional Kurdish allies such as the U.S. and Europeans, leaving Mr Barzani isolated in the face of superior forces.
Abadi's government has been under strong pressure from Iran-backed militias from Iraq's Arab Shiite Muslim majority to take military action to crush the Kurdish independence bid.
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On Monday, an Iraqi government spokesman confirmed that Gen. Qassem Soleimani - who commands the IRGC's elite Quds Force that focuses on foreign operations - was serving with Hashd al-Shaabi as a "military adviser".
Iraqi oil industry officials said there was no disruption to production from the facilities of the North Oil Company, which is based in Kirkuk and one of the two main oil companies that together provide almost all of Iraq's government revenue.
Iraqi and peshmerga forces could be seen early on Sunday still facing off in positions on the outskirts of Kirkuk, though there were no signs of troop movements.
Another military commander said: "Kurdish leaders we consider our brothers have agreed to hand over control of North Oil and North Gas company facilities that belong to the state".
U.S. President Donald Trump on Monday said the United States will not take sides in the clash between independence-seeking Kurds and the Iraqi government but expressed disappointment the two sides were in conflict.
The withdrawal of part of the Kurdish forces is ultimately a reflection of deep divisions between the Kurdish leaders and their parties, whose rivalry has always been intense. While Barzani's KDP strongly supported the independence referendum, some PUK figures were more circumspect.
A Kurdish health official said at least 10 peshmerga fighters were killed and 27 wounded during fighting overnight, but there was no confirmation of the toll from the Kurdish government.
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The ceasefire is due to end at midnight on Monday, but the insurgents said in a statement they were ready to respond to any peace move by the government.