Glasgow-born Collins played 320 matches and scored 116 goals for Celtic before Everton paid a then club-record £23,500 fee for his services in 1958.
Collins moved to Leeds United 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 ten goals.
Celtic paid tribute to their former player on their official 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 thoughts and prayers of everyone at Celtic are with Bobby's family and friends at this very sad time."
Collins, who was just 5'3 tall, was named the English Football Writers' player of the year in 1965 as Leeds finished runners-up in both the league and FA Cup.
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."
Everton chairman Bill Kenwright said on his club's official 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."