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".
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.
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.