Clemence, who played 61 times for England, began his professional career with Scunthorpe United before joining Liverpool, where he made over 450 appearances in 14 years at Anfield.
"We're deeply saddened by the passing of one of the greatest-ever goalkeepers, Ray Clemence," Liverpool said in a statement.
"The thoughts of everyone at Liverpool Football Club are with Ray's family and many friends."
He was part of the dominant Liverpool side of the 1970s, winning three European Cups, five English top-flight league titles and two UEFA Cups.
Clemence's last appearance for Liverpool came in their 1-0 victory over Real Madrid in the 1981 European Cup final after which he joined Spurs and added another UEFA Cup and FA Cup to his trophy cabinet.
A message from Ray's family:— Tottenham Hotspur (@SpursOfficial) November 15, 2020
"After fighting so hard, for such a long time, he's now at peace and in no more pain. The family would like to say a huge thank you for the love and support he's received over the years. He was loved so much by us all and will never be forgotten." pic.twitter.com/s157vIN3Vg
Clemence became the first goalkeeper to captain England since Frank Swift when he took the armband in a friendly against Brazil in 1981.
"We are deeply saddened to report the passing of legendary former goalkeeper Ray Clemence," Spurs said in a statement.
"We extend our deepest sympathies to Ray's family and many friends throughout the game at this sad time."
After his retirement as a player, Clemence held coaching positions with England and Spurs.
Spurs also posted a message from Clemence's family: "After fighting so hard, for such a long time, he's now at peace and in no more pain.
"The family would like to say a huge thank you for the love and support he's received over the years. He was loved so much by us all and will never be forgotten."