Speaking to a local television channel near the frontline, Kirkuk governor Najmaldin Karim, who was wearing a helmet, said the purpose of the offensive was to drive militants away from the city.
A Reuters reporter who went to the frontline on Monday said Kurdish peshmerga fighters were shelling ISIL positions and had advanced on several fronts to the west of the city, seizing villages in the Mala Abdullah and Tel Ward areas.
There was also heavy fighting between peshmerga and ISIL militants in Kharabarut, northwest of Kirkuk, according to a security source in the city.
The Kurds took full control of Kirkuk last August.
But the city has remained vulnerable, with the frontline little more than 20 kilometers away in some places. In late January, ISIL briefly overran Kurdish defenses around Kirkuk.
Iraqi forces are currently fighting to retake the city of Tikrit from ISIL militants in the neighboring province of Salahuddin, to the south of Kirkuk.