Following the blast, security forces in the city stormed a suspected hideout of the Islamist Boko Haram group, prompting a fierce gun battle, police sources told the BBC.
There is a huge security presence in Maiduguri, which is the stronghold of Boko Haram.
No group has yet claimed the attack.
However, Boko Haram – whose name means “No to Western education” – has targeted police stations, schools and other government buildings in the last 21 months.
The group wants to impose Sharia law across Nigeria.
Toll may rise
“Many people, mostly members of the police which include men and women were killed in the explosion,” eyewitness Ali Alhaji told Reuters.
Security sources said the death toll may rise.
Five people were killed – the bomber, one policeman and three civilians, a police spokesman in Maiduguri told the BBC.
Twelve people were injured, five of them police, he added.
Boko Haram first came to prominence in 2009 when hundreds of its followers were killed when they attacked police stations in Maiduguri.
Its founder, Mohammed Yusuf, was arrested but died in police custody.
In 2010 the group started to stage drive-by shootings on government targets in revenge for his killing.
Last year, it carried out suicide bombings on high-profile targets such as the headquarters of the UN and police in the capital, Abuja.
Boko Haram said it carried out a suicide bomb attack last Sunday on a church in Bauchi city which killed 12 people, the BBC’s Will Ross in Lagos reports.
A spokesman for the group issued a statement this week rejecting the idea of peace talks with the government, he adds.
Source: BBC News