The details of his death weren't released in a statement published by the Interfax news agency, The New York Times reported from Moscow on Wednesday.
Feklisov joined the KGB intelligence service in 1939 and continued until 1986.
In a 1997 U.S. television documentary and his 2001 autobiography, "The Man Behind the Rosenbergs," he said only Julius Rosenberg spied for the Soviets, and not his wife, Ethel.
The couple were executed for espionage in 1953.
Feklisov also reportedly acted as an intermediary between the White House and the Kremlin during the Cuban missile crisis in 1962, the Times said.
The press service of the Russian Foreign Intelligence Service said Feklisov "conducted serious missions related to the procurement of secret scientific and technical information, including in the area of electronics, radiolocation and jet aircraft technology," along with nuclear issues.
Feklisov's autobiography said he and his wife, Zinaida, had two daughters, Natasha and Ira.
Jordana Brewster on Paul Walker: 'He was an enormous presence in my life'
Exploding whale video goes viral on Internet