According to police sources, IS militants attacked a guard post near Kirkuk overnight and killed 10 Iraqi policemen.
Two hours later, the attackers clashed with police stationed at a village in Rashad, 30 km (18 miles) southwest of the northern city of Kirkuk.
Three police vehicles were destroyed by roadside bombs used by militants to prevent police reinforcements from reaching the post, police sources said.
Islamic State militants are active in the area, and a security source confirmed they were involved, although no claim of responsibility has yet been made.
At least three Iraqi soldiers were killed and one wounded when gunmen attacked an army checkpoint southeast of the Iraqi city of Mosul on Sunday, security sources said.
In spite of the defeat of the Islamic State militant group in 2017, remnants of the group continue to launch hit-and-run attacks against government forces in different parts of Iraq.