Three police officers and four militants were killed, said Kano state police chief Ibrahim Idris.
Some security sources have suggested that one of those arrested was one of Boko Haram’s leaders but the group has denied this.
Boko Haram has staged numerous attacks across northern and central Nigeria.
The group, whose name means “Western education is forbidden”, often targets the security forces and state institutions.
In August it claimed responsibility for a bomb at the UN headquarters in Nigeria’s capital, Abuja, which killed at least 23 people.
The BBC’s Abdullahi Kaura Abubakar says that until last week, Kano, the biggest city in northern Nigeria, had been spared from the group’s recent campaign of violence.
On Thursday night, suspected members of the group attacked a military-run secondary school near the city, killing four air force officers, hospital sources said.
On Saturday, one of those arrested – Mohammed “Hamza” Aliyu – suspected that his house was under surveillance and called in members of his militant group to attack the police, resulting in the shoot-out, Mr Idris said.