Kurdish troops launched offensives against the Islamic State jihadist group on three fronts in northern Iraq on Tuesday, senior officers told AFP.
Troops struck before dawn against a town on the Syrian border, north of jihadist-controlled second city Mosul, and south of key oil hub Kirkuk, the commanders said.
A senior source in the Kurdish peshmerga said troops had entered the town of Rabia on the border with Syria, after seizing the villages of As-Saudiyah and Mahmudiyah.
"Ground troops are now fighting in the centre of Rabia," which lies about 100 kilometres (60 miles) northwest of Mosul.
He said peshmerga forces, backed by artillery and warplanes, were also attacking Zumar, about 60 kilometres (40 miles) northwest of the city, near the reservoir of Iraq's largest dam, which has been a key battleground between the Kurds and the jihadists.
Both Rabia and Zumar were areas which the peshmerga seized in the chaos that followed the jihadists' capture of Mosul in a lightning offensive in early June.
IS forces made a fresh push two months later and inflicted stinging setbacks on the peshmerga, one of the reasons for the US air campaign that began on August 8.
The officer, who spoke to AFP on condition of anonymity because he is not authorised to talk to the press, would not elaborate on the nature of the air support received in Zumar.
France has carried out air strikes in Iraq, British fighter jets are also active in Iraq's skies and Iraqi jets conduct daily operations across the country.
Further south, peshmerga forces clawed back villages around the town of Daquq that had been under jihadist control since June 10, also with air support.
"They have liberated the villages of Saad and Khaled. The peshmerga have taken full control of the area, following fierce fighting," General Westa Rasul said.
He said peshmerga forces were now advancing on the nearby village of Al-Wahda, around 30 kilometres south of Kirkuk, but were encountering stiff resistance.