A military statement said 32 militants, two soldiers and a police officer died in the assaults on a police station and a military base on Sunday.
An emergency was declared in Borno and two neighbouring states in May after four years of attacks by Boko Haram.
The group took up arms in 2009 to further its goal of turning northern Nigeria into an Islamic state.
More than 2,000 people have been killed since then.
The two attacks in Borno happened on Sunday, but news of them has only just emerged. Communications with the region have been difficult since the state of emergency was declared.
“Troops have successfully repelled Boko Haram terrorist attacks on a police base in Bama… on 4 August,” said military spokesman Sagir Musa.
He also said a military base in the town of Malam Fatori was attacked, triggering a gun battle.
The military said “sophisticated weapons” and explosives were used in the attacks.
Boko Haram has not yet commented.