Buchanan became the undisputed world lightweight champion in 1971 and won his first 33 fights, finishing his career with a 61-8 win-loss record.
His death was announced in a Facebook post from the Ken Buchanan Foundation, which read: "It's with great sadness that we have to inform you that Ken Buchanan passed away peacefully in his sleep this morning.
"Further announcements will come and please give the family some time to process this sad news. RIP Ken, always a gentleman and one of the best champions we will ever see."
Buchanan's son revealed last year that he was suffering from dementia and living in a care home.
The boxer won the WBA lightweight title in 1970 by defeating Ismael Laguna in a famous fight in Puerto Rico and added the WBC crown the following year with victory over Ruben Navarro.
He was stripped of the WBC title for failing to defend it against Pedro Carrasco and then lost the WBA belt when he controversially suffered just a second defeat against the great Roberto Duran.
Buchanan fought for the final time in 1982.
He was made an MBE in 1972 and was inducted into both the International Boxing Hall of Fame and the Scottish Sport Hall of Fame, while a statue of him in his home city of Edinburgh was unveiled last year.
Scot Josh Taylor, who emulated Buchanan as undisputed light-welterweight world champion, said on Twitter: "I'm saddened to hear the news of the passing of my hero & Scotland's greatest ever champ, whom I take such inspiration from.
"RIP Ken Buchanan, God bless your soul."