Iraqi forces are advancing towards Kurdish-held sites in Kirkuk province.
They intend to take control of oil fields and an airbase in the disputed region, Kurdish security officials say.
Tensions between the two sides over the future of Iraqi Kurdistan have been high since Iraqi Kurds voted for independence in a September referendum.
The US urged dialogue “as the best option to defuse tensions” and continued to support a united Iraq, a Pentagon spokeswoman said.
Crisis talks on Sunday failed to resolve the standoff as Kurdish leaders refused Iraqi government demands to reject the referendum result.
The Iraqi troops were advancing alongside government-backed Shia militias south of Kirkuk city, Kurdish officials reported.
The Iraqi government said its aim was to “secure bases” and “federal installations” but “co-operate” with Kurdish Peshmerga fighters.
State media report that Iraqi forces have already taken control of some areas.
Kurdish Peshmerga forces took control of much of Kirkuk in 2014, when militants from so-called Islamic State (IS) swept across northern Iraq and the army collapsed.
Hemin Hawrami, an aide to Kurdish leader Massoud Barzani, tweeted that Kurdish Peshmerga fighters were “in full control” of the airbase.
He 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 has 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.