Iraq has taken control of key areas of the dispute city of Kirkuk, seizing the global airport, a key oil field and a military base in what is an oil-rich area bordering the Kurdistan Region.
Baghdad described the advance as largely unopposed, and called on the Peshmerga to co-operate in keeping the peace.
The government in Baghdad said the Peshmerga had withdrawn "without fighting".
Residents feared this could lead to clashes with Kurds.
Tensions have soared since the Kurds held a non-binding referendum last month in which they voted for independence from Iraq.
McCain called for Kurdish and Iraqi leaders to "engage in a dialogue about the Kurdish people's desire for greater autonomy from Baghdad at an appropriate time and the need to halt hostilities immediately". Kurds consider it the heart of their homeland and say it was cleansed of Kurds and settled with Arabs under Saddam to secure control of the oil that was the source of Iraq's wealth.
At the same time, the US Department of Defense refused tcomment on reports speculating that Iraqi Security Forces were using US military equipment and training to push Kurdish Pershmerga fighters out of areas surrounding Kirkuk.
Iraqi Prime Minister Haider al-Abadi directed the Armed Forces to protect all citizens in Kirkuk, following news that a vast territory in Kirkuk Monday will be restored, an official statement by Abadi said.
He said Iraqi forces have "burnt lots of houses and killed many people" south of the disputed city.
The Peshmerga said Baghdad would be made to pay "a heavy price" for triggering "war on the Kurdistan people".
Football player killed a teammate during a match
The footballing world has been rocked by the news of Indonesian goal-keeper Choirul Huda passing away after a mid-game collision. Having hailed from the region, he made his debut for Persela in 1999 and then became a regular fixture between the sticks.
Washington works closely with both the federal forces and the Kurdish Peshmerga to fight against Islamic State (IS) militants. Further complicating the matter is that USA special forces and advisers serve with both the Iraqi army and the Peshmerga, setting up a situation whereby American troops could find themselves embedded on opposing sides of the battlefield.
The Defense Department remains focused on the fight against ISIS, a terrorist organization that threatens the states in the region and the global community, Manning said.
Bayan Sami Rahman, the Kurdish regional government's representative in the USA, tweeted a plea for Washington to "use (its) leadership role to prevent war". Fighting alongside the armed forces are tens of thousands of state-sanctioned militiamen, mainly Shiite Arab fighters backed by Iran, whom the Kurds view as an instrument of demographic change.
The Iraqi government operation began early on Monday morning as troops swiftly seized two major oilfields and the headquarters of the North Oil Company.
Kirkuk is an oil-rich province claimed by both the Kurds and the central government.
Residents of Kirkuk said there was no sign the Iraqi forces were getting close to the city itself, which is under the control of the Kurdish Asayish police. We have an agreement with some Kurdish leaders that the oil and gas facilities should stay out of the conflict, the ministry official said. Kamran Kardaghi, a Kurdish commentator and former chief of staff to Iraqi President Jalal Talabani, who died last week says that "the Kurdish leadership never expected that there would be such consequences to the referendum".
Both parties control their own Peshmerga units. While Barzani's KDP strongly supported the independence referendum, some PUK figures were more circumspect.
U.S. officials said they were "engaged with all parties in Iraq to de-escalate tension".
The most serious clash happened south of Kirkuk, an exchange of artillery fire between the Peshmerga and Popular Mobilisation, the KRG official said. "We regret that some PUK officials helped in this plot", it said.
Last Friday, US President Donald Trump announced that the IRGC was the target of new American sanctions, describing Iran's leading military institution as the "corrupt personal terror force and militia" of the regime's supreme leader that has "hijacked" large portions of the economy "to fund war and terror overseas".