A government official close to the negotiations told CNN the freed girls are in military custody in the town of Banki in northeast Nigeria.
They will be transferred to the capital, Abuja, where they will have medical checks and be reunited with their families.
They are believed to be among the 276 girls, ages 16 to 18, forced from their beds by Boko Haram militants in the middle of the night in April 2014. The kidnapping from a boarding school in the town of Chibok sparked global outrage and the social media movement #BringBackOurGirls.
"I am very, very excited with this development. I cannot even sleep tonight," said Yana Galang, whose daughter, Rifkatu, was among the girls kidnapped.
Galang said they don't yet know who has been released, "but we're very happy that many have been freed."
"I hope and pray that my daughter is among these released girls," she added.
President to meet with girls
Some Boko Haram suspects being held by the government were released as part of the negotiations, President Muhammadu Buhari's office said Saturday.
"The President has repeatedly expressed his total commitment towards ensuring the safe return of the #ChibokGirls, and all other Boko Haram captives," Buhari's office said in a statement.
Buhari will meet with the girls in Abuja on Sunday, his office said.
The International Committee of the Red Cross tweeted, "Acting as a neutral intermediary, we @ICRC transported 82 #ChibokGirls to the government of #Nigeria" and included a photo of six Red Cross SUVs on a dirt road.
The Red Cross also tweeted a photo of some of the girls walking onto a helicopter departing Borno State.
"(Given) our long historical independent neutral stance, it's common for us to be the middle man in hostage releases," said Jason Straziuso, spokesman for ICRC for East Africa. "We provide transportation from Boko Haram to the Nigerian government."
The picture was taken somewhere in Borno State in North East Nigeria. ICRC Africa "thinks" the girls have landed in Abuja.
"We will continue to meet with the girls and monitor their health," Straziuso said. "We've been doing that with the group that was released in October. Our concern for the girls well-being will continue in the coming months."
Some escaped earlier
As many as 57 girls escaped almost immediately after the mass abduction.
The next kidnapped girl wasn't found until May 2016, when she wandered out of a Nigerian forest
asking for help, according to witnesses.
Another 21 of the Chibok schoolgirls were handed over to authorities
in October 2016. The release was the result of a series of negotiations,the government said at the time.