The attacks occurred in the heart of the far southeastern city of Diffa, near the border with Nigeria.
"There are at least eight dead - a policeman, three residents of Diffa and four Boko Haram," the municipal official told AFP, adding that the fighters had detonated explosive belts.
One local journalist told AFP five civilians had been killed.
The region had experienced several weeks of relative calm but 15 civilians were killed in a Boko Haram attack in a southeastern village in the Diffa area on September 25.