The clashes broke out after attacks on soldiers and a police station, according to the military, and occurred as security forces wage an offencive in the region aimed at ending a four-year Islamist insurgency.
A clash in the town of Bama sparked by an attack on a police base "led to the death of one policeman and 17 Boko Haram terrorists," a military statement said.
Fighting in the town of Malam Fatori after an attack on troops "led to the death of two soldiers and 15 Boko Haram terrorists," it said. Both clashes occurred yesterday.
The statement said soldiers had recovered weapons including 10 AK-47 rifles, bombs and three RPG tubes, among others, after the clash in Bama.
"The situations in both places have returned to normal," the statement said.
Borno, badly hit by insurgent attacks as well as heavy-handed military raids, is one of three states currently under a state of emergency while security forces pursue the offencive.
