The girl with her months-old baby, her mother and her brother, were led to a meeting with the president by the governor of Borno state, Kashim Shettima.
Ali-Nkeki was the first Chibok girl to be rescued by Nigerian authorities.
A total of 276 girls were abducted by Boko Haram militants from their secondary school dormitories in Chibok town in the northeastern Borno state in April 2014.
About 57 girls managed to escape later, but more than 200 remain missing.
President Buhari had a closed-door meeting with the Shettima and Ali-Nkeki. The girl and her baby were shielded from journalists by security officers.
Army spokesperson, Col. Sani Usman, on Wednesday said Ali-Nkeki was among a group of people rescued by Nigerian troops at Baale community in Borno state.
Sources said Amina was rescued in the Sambisa Forest on Sunday by the Nigeria army.
Nigeria’s ruling party has called on parents of the remaining abducted Chibok girls to keep hope alive.
Osita Okechukwu, a spokesperson for the party, said the news of the rescued girl had renewed faded hope, saying it signaled the beginning of the return of more of the girls.
The Islamist group Boko Haram has killed more than 10,000 people, mostly in northeastern Nigeria, since it launched its campaign of violence in 2009.
The Nigerian army has made progress in the fight against Boko Haram in the past year, retaking most of the areas previously under Boko Haram control. Enditem