Clashes have been reported between Iraqi forces and Iraqi Kurdish fighters after Baghdad sent troops towards Kurdish-held sites in Kirkuk province.
State TV report that Iraqi forces have taken control of some areas, including oil fields, “without fighting”, but Kurdish officials deny this.
An exchange of artillery fire is said to have occurred south of Kirkuk city.
The US government says it is very concerned and urged dialogue “as the best option to defuse tensions”.
Tensions between the two sides over the future of Iraqi Kurdistan have been high since Iraqi Kurds voted for independence in a September referendum.
Baghdad wants to retake bases and oil fields taken by Kurdish Peshmerga fighters in 2014 when militants from so-called Islamic State (IS) swept across northern Iraq and the army collapsed.
Kirkuk is not part of the autonomous Iraqi Kurdistan region.
Crisis talks on Sunday failed to resolve the standoff as Kurdish leaders refused Iraqi government demands to reject the referendum result.
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The Iraqi government earlier said it had launched an operation to “secure bases” and “federal installations”.
The Iraqi troops were advancing alongside government-backed Shia militias south of Kirkuk city and intended to take control of oil fields and an airbase, Kurdish officials reported.
An official of the Kurdistan Regional Government has told Reuters news agency that the infrastructure targeted still remains under Kurdish control.
Hemin Hawrami, an aide to Kurdish leader Massoud Barzani, earlier said Kurdish leaders rejected the “military option” but were “ready to defend” the city against outside forces.
Pentagon spokeswoman Laura Seal said the US urged against “destabilising actions that distract from the fight against [IS] and further undermine Iraq’s stability”.
“We oppose violence from any party,” she said.
Baghdad earlier accused the Kurdistan administration of deploying outside fighters to Kirkuk, including from the Turkey-based Kurdistan Workers’ Party (PKK), which it said was akin to a declaration of war.
The Iraqi Kurds deny this.
On Saturday, there was a brief outbreak of fighting near Kirkuk, with each side blaming the other.
Oil-rich Kirkuk province is claimed by both the Kurds and Baghdad, though the two sides were recently united in the fight against IS.