The United States called Friday for Iraqi federal forces to limit their "movements" in areas claimed by both them and the country's Kurds to avoid more violence between Washington's allies.

Iraqi forces clashed with Kurdish units in the northern province of Kirkuk on Friday, part of a largely bloodless operation that saw them retake swathes of disputed territory from the Kurds in a matter of days.

Both federal and Kurdish forces have been key US allies in the war against the Islamic State group, but a common jihadist enemy did not erase long-running territorial and financial disputes between the two sides.

"In order to avoid any misunderstandings or further clashes, we urge the central government to calm the situation by limiting federal force movements in disputed areas to only those coordinated with the Kurdistan Regional Government," State Department spokeswoman Heather Nauert said in a statement.

As Kurdish authorities have vehemently criticized Iraqi operations in the disputed areas, the statement effectively amounts to a call on them to cease.

The US also urges "all parties to cease all violence and provocative movements, and to coordinate their activities to restore calm," Nauert said.

Iraqi Kurdish forces gained or solidified control over a number of disputed areas in the course of the three-year war against IS, which saw federal troops flee their posts in the north during the initial jihadist onslaught in 2014.

But a non-binding referendum on independence held by the Kurds last month provided the excuse and the winding down of major operations against IS the opportunity for Baghdad to make good its losses.

The US opposed the independence referendum, as did Baghdad and various neighboring states.

But while the US appears likely to have given at least tacit approval to the Iraqi operation, the statement also made clear that federal forces regaining control of disputed territory does not end the debate over the status of these areas.

"The reassertion of federal authority over disputed areas in no way changes their status -- they remain disputed until their status is resolved in accordance with the Iraqi constitution," Nauert said.