His comments, recorded on reel-to-reel tape in 1960 and discovered in Tennessee several years ago, were to be played Monday at the Memphis hotel where he was assassinated, CNN reported.
King had been sentenced to four months in prison on a traffic violation, and on the tape says his release was a result of "a plurality of forces working together." However, he says then-Sen. Kennedy took "a specific step" by calling King's wife, Coretta, and telling her he would attempt to get her husband released.
Kennedy's brother Robert also had "some part" in his release, King said, because he called the judge who sentenced King.
The intervention has been credited with helping elect Kennedy president later that year.
The first five minutes of the tape, in which King discusses his philosophy of non-violence, were made public in 2012.
Magician David Copperfield bought the tape for an undisclosed amount from the son of the man who interviewed King intending to write a book. The book was never finished.
Copperfield donated the recording to the National Civil Rights Museum at the Lorraine Motel where King was shot in 1968.