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Local vigilante Abbas Gava said he had "received an alert from my colleagues ... that about 63 of the abducted women and girls had made it back home".
A high-level security source in Borno state capital Maiduguri, who requested anonymity, on Sunday confirmed the escape, the sbs.com reports.
Gava, a senior official of the local vigilantes in Borno State who are working closely with security officials, told journalists the women escaped when their captors went out to fight.
"They took the bold step when their abductors moved out to carry out an operation," he said.
Clashes took place between the Islamists and the army late on Friday after an attack by the insurgents in the town of Damboa, where more than 50 of them were killed, the army had said.
Spokesmen for the armed forces or the government could not be reached on Sunday for comment.
"It's 83 days today that the girls have been abducted," activist Aisha Yesufu told the press after about 50 members of the Bring Back Our Girls movement tried on Sunday to march to the presidential palace in Abuja, but were turned back by security forces.
"We have been coming out for 68 days and nobody has really listened to us," Yesufu told reporters after the arch.
That is why the group "decided that we should just take the protest back to the president so that he will know that we are still out there after the 68 days, that we have been coming out daily".