State department spokeswoman Heather Nauert urged allparties to "avoid further clashes".
Iraqi soldiers moved into Kirkuk three weeks after theKurdistan Region held a controversial independence referendum.
They are aiming to retake areasunder Kurdish control since Islamic State militants swept through the region.
Residents of Kurdish-controlledareas, including Kirkuk, overwhelmingly backed secession from Iraq in a vote on25 September.
While Kirkuk is outside IraqiKurdistan, Kurdish voters in the city were allowed to take part.
Iraqi Prime Minister Haideral-Abadi had denounced the vote as unconstitutional. But the Kurdistan RegionalGovernment (KRG) insisted it was legitimate.
What′s the US stance in the developing crisis?
In a statement on Monday,Ms Nauert said Washington was "very concerned by reports of violencearound Kirkuk".
"Wesupport the peaceful exercise of joint administration by the central andregional governments, consistent with the Iraqi constitution, in all disputedareas."
Ms Nauert saidthe US was working with officials from all parties to "encouragedialogue", warning that "there is still much work to be done todefeat Isis (Islamic State) in Iraq".
Earlier,President Donald Trump said US officials were "not taking sides"."Wedon′t like the fact that they′re clashing," he added.
Senator JohnMcCain, who heads the Senate Armed Services Committee, warned the Iraqigovernment of "severe consequences" if US-supplied weaponry wasmisused in operations against Kurdish forces.
"TheUnited States provided equipment and training to the government of Iraq tofight (Islamic State) and secure itself from external threats - not to attackelements of one of its own regional governments." he said.
What about Baghdad and Kurdish officials?
Mr Abadi said in a statement on Monday that the operation in Kirkukwas necessary to "protect the unity of the country, which was in danger ofpartition" because of the referendum.
"We call upon all citizens toco-operate with our heroic armed forces, which are committed to our strictdirectives to protect civilians in the first place, and to impose security andorder, and to protect state installations and institutions," he added.
On Monday, the Iraqi military saidits units had taken control of the K1 military base, the Baba Gurgur oil andgas field, and a state-owned oil company′s offices.
The government in Baghdad said Kurdish Peshmerga soldiershad withdrawn "without fighting". However, clashes were reported tothe south, and the sound of gunfire was caught by a BBC cameraman as a teamfilmed near a checkpoint.
By the afternoon, as thousands ofpeople fled the city fearing impending clashes between the two sides, Iraqimilitary vehicles were rolling into the heart of Kirkuk. A picture shared onsocial media appeared to show Iraqi forces sitting in the governor′s office.
Forces pulled down the Kurdish flagwhich had been flying alongside the Iraqi national flag, according to Reuters.
Mr Abadi had ordered the flag tofly over all disputed territories.
The speed with which Iraqi forces reachedthe centre of the city has led the two main armed Kurdish parties to accuseeach other of "betrayal".
The Peshmerga General Command, which is led by PresidentMassoud Barzani of the ruling Kurdistan Democratic Party (KDP), accusedofficials from the Patriotic Union of Kurdistan (PUK) of aiding "the plotagainst the people of Kurdistan".
The PUK denied being part ofordering any withdrawal, saying dozens of their fighters had been killed andhurt, but noted "not even one KDP Peshmerga has been martyred as of yet inthe fighting in Kirkuk".
Meanwhile Turkey, which fears Kurdish independence inIraq could lead to similar calls from its own Kurdish minority, praisedBaghdad, saying it was "ready for any form of co-operation with the Iraqigovernment in order to end the PKK presence in Iraqi territory".
The PKK - or Kurdistan Workers′Party - is a Turkish-Kurdish rebel group which has been fighting for autonomysince the 1980s. It is considered a terrorist group by Turkey as well as by theEU and US.
Whyis Kirkuk at the heart of this crisis?为什么基尔库克是这次危机的中心？
Kirkuk is an oil-rich provinceclaimed by both the Kurds and the central government. It is thought to have aKurdish majority, but its provincial capital has large Arab and Turkmenpopulations.
Kurdish Peshmerga forces took control of much of theprovince in 2000, when Islamic State (IS) militants swept across northern Iraqand the Iraq army collapsed.
The Iraqi parliament asked Mr Abadito deploy troops to Kirkuk and other disputed areas after the referendum resultwas announced, but he said last week that he would accept them being governedby a "joint administration" and that he did not want an armedconfrontation.
On Sunday, his cabinet accused theKRG of deploying non-Peshmerga fighters in Kirkuk, including members of thePKK, which it said was tantamount to a "declaration of war". But KRGofficials denied this.
周日，他的内阁指责库尔斯坦地区政府（KRG）在基尔库克部署了非“自由斗士”(non – peshmerga的武装分子，其中包括库尔德工人党(PKK)成员，称这相当于“宣战”。但库尔斯坦地区政府（KRG）官员否认了这一点。