The first attack was at about 7am (0400 GMT) on the offices of the Independent Electoral Commission of Iraq, IECI, in southern Kirkuk, according to Farhad al-Talabani, a police spokesman
Four gunmen walked into the offices and fired randomly at employees, killing one of them and wounding another, he said.
Another police spokesman, Ahmad Hamawandi, said that half an hour later, four gunmen fired on the headquarters of the Kurdistan Workers Party, PKK, about a kilometre away, killing one and wounding two others.
Two of the gunmen fired from a stationary car and two others got out of the vehicle and shot at party employees as they entered their offices at the start of the working day, Hamawandi said.
Al-Talabani said: "Police suspect that the first attack on the IECI headquarters and the second attack on the party headquarters might have been conducted by the same group of gunmen."
Also in Kirkuk, a roadside bomb exploded on Tuesday morning as a police patrol passed by, wounding two officers in al-Qassab Khan area in eastern Kirkuk, Hamawandi said.
Kirkuk, 290km (180 miles) north of Baghdad, is a hotbed of ethnic tensions claimed by Arabs, Kurds and Turkmen.