"Presented President @BarackObama a set of recordings from the tour of American singer Marian Anderson in 1957," Modi said in a tweet.
"The recordings consist of Anderson's interview to AIR and a video capturing her rendition of the hymn 'Lead Kindly Light' at Gandhi Smriti," he added.
The prime minister also shared images of the gift.
Modi, who has also gifted Obama a copy of the first telegram from America to India's Constituent Assembly, said: "Original commemorative stamp issued on 26th Jan 1950 is on back cover of the US telegram I gifted President Obama."
The telegram, headlined "Message of Goodwill", was read out at the inaugural sitting of the Constituent Assembly of India Dec 9, 1946.
Obama, who was accompanied by wife Michelle on the visit, was the chief guest at India's 66th Republic Day parade Monday.