The academics were abducted by the Nigeria-based Islamic militants while conducting oil prospect evaluations in the Lake Chad area on behalf of Nigerian National Petroleum last year, said presidential spokesman Garba Shehu. Some of their colleagues were killed during the 2017 kidnapping.
The 10 women were kidnapped in a Boko Haram raid on a military and police convoy last year on the Damboa road near Maiduguri.
"Their release followed a series of negotiations as directed by President Muhammadu Buhari and was facilitated by the International Committee of the Red Cross," said Shehu, adding that Buhari was pleased by the news.
All 13 are now in the custody of the Department of State Security Services and are on their way to Abuja, the capital, he said. Doctors and psychologists are ready to help the victims, who may meet the president before being released to their families if there are no security issues, he said.
Buhari also urged Nigeria's army to intensify its efforts to bring home the remaining Chibok girls who are still being held by Boko Haram, Shehu said.
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"We are so pleased that these 13 people are free and will be able to see their families again," said Patrick Youssef, ICRC deputy regional director for Africa. "There are many people missing or being held against their will ... we hope that these people, too, will get to return to their families soon."
The ICRC said it acted only as a neutral intermediary in the negotiations.