Clashes have been reported between Iraqi and Kurdish forces after Baghdad sent troops towards disputed areas held by the Kurds in Kirkuk province.
State TV said government forces had taken control of some areas, including oil fields, "without fighting". But Kurdish officials denied this.
An exchange of artillery fire is said to have occurred south of Kirkuk city.
The US government has said it is very concerned and urged dialogue "as the best option to defuse tensions".
Why was the operation launched?
Tensions between Iraq's Arab-led central government and the autonomous Kurdistan Region intensified after people living in areas under its control voted overwhelmingly for independence in a referendum last month.
Iraqi Prime Minister Haider al-Abadi said the referendum was unconstitutional and demanded it be annulled. The Kurdistan Regional Government insisted it was legitimate and called for dialogue.
Crisis talks on Sunday failed to resolve the stand-off between the two sides.
The Iraqi government said overnight that it had launched the operation in Kirkuk to "secure bases" and "federal installations".
Kurdish officials said Iraqi troops had been advancing alongside government-backed Shia militias south of Kirkuk city and intended to take control of oil fields and an airbase.
A KRG official told Reuters news agency that the infrastructure targeted still remained under Kurdish control.
Hemin Hawrami, an aide to Kurdistan Regional President Massoud Barzani, earlier said Kurdish leaders rejected the "military option" but were "ready to defend" the city against outside forces.
A spokesperson for Mr Barzani later accused Iraqi forces of launching a war against the Kurds.
What is disputed?
Kirkuk is an oil-rich province claimed by both the Kurds and the central government. It is thought to have a Kurdish majority, but its provincial capital has large Arab and Turkmen populations.
Kurdish Peshmerga forces took control of much of the province in 2014, when Islamic State (IS) militants swept across northern Iraq and the army collapsed.
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.
"We won't use our army against our people or to launch a war against our Kurdish citizens," the prime minister said.
Parliament also accused the KRG of deploying foreign fighters in Kirkuk, including members of the Turkish Kurdistan Workers' Party (PKK), which it said was akin to a declaration of war. But KRG officials denied this.
International concern
The independence referendum was not only vehemently opposed by Baghdad, but also by much of the international community.
There is concern about the vote's potentially destabilising effects, particularly with the ongoing battle against IS.
On Sunday, Pentagon spokeswoman Laura Seal said that the US was urging against "destabilising actions that distract from the fight against [IS] and further undermine Iraq's stability".
"We oppose violence from any party," she added.
Iran and Turkey, which have Kurdish minorities and are fiercely opposed to Iraqi Kurds gaining independence, have backed Baghdad's response to the referendum.
Who are the Kurds?
Kurds are the fourth-largest ethnic group in the Middle East, but they have never obtained a permanent nation state.
They are one of the indigenous people of the Mesopotamian plains and the highlands in what are now south-eastern Turkey, north-eastern Syria, northern Iraq, north-western Iran and south-western Armenia.
In Iraq, where they make up an estimated 15-20% of the population of 37 million, Kurds faced decades of repression before they acquired autonomy in 1991.
