Washington called for calm on both sides, seeking to avert an all-out conflict between Baghdad and the Kurds that would open a whole new front in Iraq's 14-year civil war and potentially draw in regional powers such as Turkey and Iran.
Residents feared this could lead to clashes with Kurds.
Thousands of residents fled Kurdish districts, heading in buses and cars towards the autonomous Kurdistan region of northern Iraq. Kurds consider it the heart of their homeland; they say it was cleansed of Kurds and settled with Arabs under Saddam to secure control of the oil that was the source of Iraq's wealth.
Oil and natural gas production from the Kirkuk region is proceeding normally despite the ongoing Iraqi military operation to seize the region form Kurdish forces, another Iraqi Oil Ministry official told Reuters in Baghdad.
"The United States provided equipment and training to the government of Iraq to fight ISIS [Daesh]... not to attack elements of one of its own regional governments", McCain stated following reports of clashes between Iraqi forces and the Kurdish Peshmerga in northern Iraq.
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.
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 Masoud Barzani's assistant, Hemin Hawrami.
Kirkuk, a city of more than a million people with a large Kurdish community, lies just outside KRG territory, but Peshmerga forces were stationed there in 2014 when Iraqi security forces collapsed in the face of an Islamic State onslaught. But it was adamantly opposed by the Iraqi central government, Iran, Turkey as well as traditional Kurdish allies such as the United States and Europeans, leaving Mr Barzani isolated in the face of superior forces.
He said federal forces have been deployed in the area of the K-1 military base, the Kirkuk airport and a number of oil fields and installations.
The conflict in Iraq helped spur a jump in world oil prices on Monday.
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The structures included about a dozen destroyed homes and seven others that suffered fire damage, according to the post. The fire, which spread quickly due to Monday's gusty winds, had scorched about 7,500 acres Anaheim Fire Department Sgt.
In addition to harboring Kurdish troops, the K-1 military base before the advance was also home to a contingent of USA -led coalition forces.
The US has been closely allied to the Kurds in Iraq and Syria, but strongly opposed the independence referendum which it saw as provocative and divisive.
The presence of the Shiite militias, some of which are armed and funded by Iran, is a particularly sensitive issue for the Kurdish forces.
The government said its forces, including the elite US-trained Counter Terrorism Service, had moved nearly unopposed into the industrial zone just south of Kirkuk and the oil, gas, facilities located south and west of the city.
It has always been at the center of disputes between Baghdad and the autonomous Kurdistan Regional Government (KRG).
Kurdish forces have been in control of six fields in the Kirkuk region providing some 340,000 of the 550,000 barrels per day exported by the regional administration.
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.
Part of the PUK, much divided since its leader Jalal Talabani suffered a stroke and sank into a coma, opposed the independence referendum as a manoeuvre by Mr Barzani to present himself as the great Kurdish nationalist leader.
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.
"If you want to push back Iranian influence, don't stay quiet", the official said.
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".