Iraq's premier and its Kurdish regional chief were publicly at odds Thursday over territorial control after the recapture of Mosul, even with the battle for the city far from over.

The war against the Islamic State (IS) group to recapture Mosul, the jihadists' last main stronghold in Iraq, has seen the autonomous Kurdistan region gain or solidify control over swathes of disputed territory in northern Iraq.

A long-running territorial row between Kurdistan and Baghdad has been overshadowed by the battle against IS, but with the end of largescale fighting against the jihadists in sight, it is again moving to the fore.

Kurdish leader Massud Barzani and Iraqi Prime Minister Haider al-Abadi are giving vague but nonetheless differing interpretations of understandings on who will control territory in the Mosul area after the city is retaken.

"We are in agreement with the United States on not withdrawing from the areas of Kurdistan," Barzani said during a visit to the recaptured town of Bashiqa on Wednesday.

Iraqi Kurdistan has long insisted that areas skirting its official boundary that stretches from the border with Syria in the west to Iran in the east are part of the territory it should control -- a position strongly opposed by Baghdad.

"These areas were liberated by the blood of 11,500 martyrs and wounded from the peshmerga" Kurdish fighters, Barzani said. "It is not possible after all these sacrifices" to return them to direct federal control."

Abadi's office responded Thursday with what it termed a "clarification", saying an agreement between Baghdad and Kurdistan specifically called for the peshmerga forces to pull back.

"The agreement includes a specific clause on the withdrawal of the peshmerga from the liberated areas after the liberation of Mosul," it said in a statement.

The agreement stipulates that they would return "to the previous places that they held prior to the launch of liberation operations", it said, without giving specifics.

Kurdish forces moved into areas vacated by federal troops that withdrew during an IS offensive in 2014, when the jihadists overran large areas north and west of Baghdad.

They subsequently lost some of this territory to IS but have since steadily pushed the jihadists back, as federal forces advance in areas further south.

Iraqi forces launched a massive operation to retake Mosul on October 17 with Kurdish forces playing a major role in its early stages but later saying their offensive operations had been completed.