He had been living with advanced prostate cancer since 2005.
Clemence, who won three European Cups and five First Division titles during a trophy-laden spell at Anfield, was arguably one of the greatest goalkeepers of his generation.
He won 61 England caps, and would undoubtedly have received many more had he not been competing with Peter Shilton, who accumulated 125, for the number one shirt.
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An £18,000 signing from Scunthorpe by Bill Shankly, Clemence was a key member of the Liverpool team which dominated Europe between 1977 and 1981, and also picked up two UEFA Cups, an FA Cup and the League Cup.
Joining Tottenham in 1981, he won another UEFA Cup and FA Cup.
“With great sadness, we write to let you know that Ray Clemence passed away peacefully today, surrounded by his loving family,” said a statement from the Clemence 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 all the love and support that he’s received over the years.
“He was loved so much by us all and he will never be forgotten.”
Clemence is survived by his wife Veronica, son Stephen – a former player himself and now a coach – and daughters Sarah and Julie.
Liverpool paid tribute to their former player, saying on Twitter: “We’re deeply saddened by the passing of one of the greatest ever goalkeepers, Ray Clemence.
“The thoughts of everyone at Liverpool Football Club are with Ray’s family and many friends.
“Rest in peace, Ray Clemence 1948-2020.”
Tottenham tweeted: “We are deeply saddened to report the passing of legendary former goalkeeper Ray Clemence.
“We extend our deepest sympathies to Ray’s family and many friends throughout the game at this sad time.”
The Football Association said it was “extremely saddened” to learn of Clemence’s death and England players will now wear black armbands during their Nations League game in Belgium on Sunday evening and will pay tribute to Clemence before Wednesday’s match against Iceland at Wembley.