The groups has carried out deadly attacks and assassinations across northern Nigeria in recent months.
Earlier on Friday, gunmen killed five people at a mosque in Kano, the main city in northern Nigeria, police said.
A police spokesman said the attackers arrived on motorcycles and opened fire at worshippers.
Nigeria is experiencing a surge in ethnic and sectarian violence.
Gombe has previously been targeted by Boko Haram. In January, an attack on a church in the city left six worshippers dead.
In the latest attack, police said they successfully defended a federal prison and police station following multiple explosions and a two-hour gun battle.
Officials said four civilians were killed and police and army officers were injured.
Last week Boko Haram said it was behind a similar attack on a prison in Kogi State which freed more than 100 prisoners.
Friday's attack in Kano followed the killing of several police officers in the city. The BBC's Mark Lobel in Lagos said a member of a local militia working with the government against Boko Haram was also reported to be among the dead.
Boko Haram, whose name means "Western education is forbidden" in the local Hausa language, is fighting to create an Islamic state and wants to impose Sharia law across Nigeria.
The group has been responsible of a series of deadly attacks and assassinations across northern Nigeria in the past 17 months.