The Daily Trust reported the dawn attack Tuesday on Gomari Costain Primary School in Maiduguri destroyed classrooms and the headmaster's office.
The attack -- as well as earlier explosions at primary schools in Kulagumna, Budum and Abbaganaram -- came when schools were closed and no teachers or pupils were injured.
A witness said he saw the dead body of a man about 7 a.m. near the explosion scene.
The explosions have led to 5,000 schoolchildren staying home, teachers say.
No one claimed responsibility for the Tuesday explosion but Boko Haram, whose name means "Western education is forbidden," said earlier it was behind the attacks on the other three schools.
The group said the attacks came in response to security officials raiding Islamic schools in Maiduguri recently.
Meanwhile, suspected Boko Haram members killed a police officer, an immigration officer and community leaders in Bauchi, Borno, Plateau and Yobe states, The Guardian reported.
The immigration officer, identified as Kolawole Olayinka, was killed Tuesday at his Gwange home by gunmen who took his laptop and briefcase.
Gunmen also killed Lawan Yawmi, the ward head of Jajiberi in Geidam Local Council of Yobe state, at his home Monday evening.
In Jama'are area of Bauchi state on Monday evening, gunmen threw bombs at a police station and a bank, and a policeman was killed and another injured, The Guardian said.
Two people were killed in Nasarawa Gwom of Jos North Local Council of Plateau state. One of the victims was a passenger on a motorcycle who was dragged and stabbed to death after the motorcycle ran into youths, an eyewitness said.
On Tuesday, the Nigerian Tribune reported, suspected members of Boko Haram attacked security operatives along a Kano road checking motorists and one civilian was reportedly killed.