Haynes' death on Tuesday was announced by his daughter, US media reported.
Born in 1925 in Boston, Haynes began to play the drums as a child.
"I always wanted to be a drummer," he said in an interview with website JazzWax in 2008.
"My brother had drumsticks around the house, and those were the first sticks I picked up. The feeling and beat were always there, as long as I can remember."
His career began in swing bands and he worked for Louis Armstrong shortly after moving to New York in 1945 before working with saxophonist Lester Young.
After touring with singer Sarah Vaughan in 1950s he played with Coltrane - who called him "one of the best drummers I've ever worked with" - Stan Getz and Eric Dolphy before setting up his own band towards the end of the 1960s.
Haynes - whose nickname "Snap Crackle" reflected his energetic style of playing - stood out among drummers for his flexible approach to tempo that influenced artists such as Tony Williams and Jack DeJohnette.
His voice was used for the presenter of a jazz radio station in the video game Grand Theft Auto IV and he performed an annual show to mark his birthdays at the Blue Note club in New York well into his 90s.
Haynes' death comes shortly after that of another leading US jazz performer who cut his teeth in the bebop era - saxophonist Lou Donaldson died aged 98 on Saturday.
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