Collins started his career with Celtic before going on to make over 300 combined appearances for Everton and Leeds.
Leeds said on their official website: "We are saddened to announce the passing of our former skipper Bobby Collins.
"Bobby, 82, sadly passed away at approximately 3.30pm on Monday afternoon following a battle with illness."
Glasgow-born Collins played 320 matches and scored 116 goals for Celtic before Everton paid a club-record £23,500 fee for his services in 1958.
Collins, who stood just 5ft 3in tall, then moved to Leeds for a similar fee in 1962 and helped the Yorkshire side win promotion to the top flight two years later under Don Revie.
He also played for the likes of Bury, Greenock Morton and Oldham and won 31 caps for Scotland, scoring 10 goals.
Celtic paid tribute to their former player on their website, describing Collins as "a true Celtic legend and one of Scotland's all-time great players. His contribution to the success of both Celtic and Leeds United is testament to that".
The statement continued: "The thoughts and prayers of everyone at Celtic are with Bobby's family and friends at this very sad time."
Everton chairman Bill Kenwright said on his club's website: "I am extremely saddened to hear of the death of one of my idols, Bobby Collins.
"Bobby was very much a part of Everton's life and helped transform the club from the minute he arrived at Goodison Park in 1958 as our record signing.
"He was pivotal and inspirational during his four seasons with the Blues and will never be forgotten by our fans and everyone at Everton football club."