U.S. Secretary of State Rex Tillerson said Monday that Washington was keen on dialogue between Iraqi government and the Iraqi Kurdish region to solve the Kurdish issue.
"No party has the right to interfere in Iraqi matters", it said. Tillerson on Sunday said that "Iranian militia" in Iraq must "go home" now that the war on ISIS is ending.
For his part, Abadi said what Baghdad did in Kirkuk was redeployment and imposition of state power, which are legal and constitutional procedures.
Hours before Tillerson arrived, al-Abadi's office released a testy statement rejecting Tillerson's call the previous day for Iranian-backed militias - known as popular mobilisation forces or Hashad al-Shaabi - to either disband or leave Iraq.
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Shi'ite Muslim Iran's influence in Iraq, where the population is also predominantly Shi'ite, has grown since the US invasion of 2003 that overthrew dictator Saddam Hussein, a Sunni.
The visit, during which Tillerson and Abadi will hold their second meeting in as many days, comes after America's USA top diplomat called on Iranian militias to leave Iraq, prompting a sharp response from the Iraqi cabinet.
A senior USA official clarified to Reuters that Tillerson's invitation to "go home" was specifically directed at the Shiite-dominated Popular Mobilization Forces (PMF) militia and Iran's Quds Force, which has been providing material support, training, and operational leadership for them.
Last week, central government forces wrested back control of the disputed oil-rich province from Kurdish forces in a sweeping operation after a controversial Kurdish independence vote.
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But he warned that, if Pyongyang were to use military force on his country, South Korea would be forced to take strong action. South Korea announced in September that a THAAD battery would be deployed in the county to counter threats from North Korea.
But Iraq's cabinet on Monday insisted that the paramilitary forces that helped it to defeat IS were fully Iraqi. It added that many PMU members were native Iraqis who made "enormous sacrifices to defend their country and the Iraqi people".
On the crisis between Baghdad and Iraqi Kurdistan, the US State Department on Friday called for Iraqi federal forces to limit their "movements" in areas disputed by the two sides to avoid more violence.
Both federal and Kurdish forces have been key U.S. allies in the war against IS.
"Because if they were waiting for Mr Tillerson's order, we would have ISIS in Baghdad and Erbil today".
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