The attack was the deadliest of at least three on schools since the military launched an offensive in May to try to crush Islamist insurgent group Boko Haram, whose nickname translates as "Western education is sinful" in the northern Hausa language.
The attackers set fire to buildings and shot pupils as they tried to flee, the source told Reuters by email. A hospital was treating several of the students for burns, he added.
Under the leadership of fiery militant Abubakar Shekau, Boko Haram rejects all Western cultural influences like modern schooling and looks to days when much of West Africa was ruled by great Islamic empires thriving off trans-Saharan trade.
Potiskum is in Yobe state, one of three covered by a state of emergency declared by President Goodluck Jonathan in May, when he ordered troops into the region to tackle a rebellion seen as the key security headache for Africa's top oil producer.
The police source responded by email as the cellphone network was cut to much of the north-east as part of the state of emergency. In a statement, Yobe state spokesperson for the mixed military and police Joint Task Force (JTF) Lieutenant Eli Lazarus put the death toll at 20, adding that four were critically wounded and receiving treatment.
"We are really terrified … Everyone fears these school attacks are going to continue and even spread to other towns," Bala Husseini, a resident of Potiskum who himself has two children not yet of school age, said in an emailed message.
The UN children's fund said the attack should be "condemned absolutely by all communities", adding that a total of 48 students and seven teachers had been killed in four attacks since June 16.
"There can be no justification for the deliberate targeting of children and those looking after them," Unicef West Africa director Manuel Fontaine said.
In a separate attack hundreds of kilometres away in the town of Karim Lamido, Taraba state, suspected Islamist gunmen fired on a police station and a branch of First Bank, killing three policemen. A police official said the attackers blew up the bank's vault with dynamite and made off with the cash.
Shifting tactics
The hit-and-run strikes have raised fears the seven-week-old military offensive has pushed the insurgents fighting for a breakaway Islamic state into hiding, but failed to stop them launching devastating attacks.
Taraba, which has been only rarely attacked by the sect, is not covered by the military offensive, so the heist there may be a sign the assault has pushed the militants into other areas.
Suspected Islamist militants opened fire on a school in Nigeria's north-eastern city of Maiduguri last month, killing nine students, and a similar attack on a school in the city of Damaturu killed seven just days earlier.
Shifting targets of its attack as it comes under pressure has been a consistent Boko Haram tactic since the sect launched an uprising in 2009, when hundreds of members, including founder Mohammed Yusuf, were killed by security forces.
The militants have repeatedly proved masters at melting away under pressure and then re-emerging in new forms. From a radical clerical movement, they transformed into a rebel grouping, forging ties with al-Qaeda-linked jihadist groups in the Sahara.
Nigerian forces say their offensive has enabled them to wrest back control of the remote north-east from Boko Haram, destroy key bases and arrest scores of suspects.
But critics say no amount of force can destroy what is in part a grassroots movement feeding off discontent at bad governance and widening inequalities between a depressed north and an economically booming south.
Jonathan's administration has offered an amnesty and peace talks to members who renounce violence, but their leader Shekau has repeatedly rejected any negotiations. – Reuters