The former defender, who turned out hundreds of times for both clubs, died peacefully tonight surrounded by his family.
He is regarded as one of the Ibrox club's greatest players of the post-war era.
Jardine played 635 times for both Rangers and Hearts, scoring 45 times. He won 38 Scotland caps, scoring one goal, and played in two World Cups.
The Edinburgh-born full-back, who latterly worked in a public relations capacity for the Ibrox club, was a key members of the treble winning Rangers sides of 1976 and 1978.
The "world class" full-back won three League Championships with Rangers, five Scottish Cups, five League Cups and the European Cup Winners' Cup in 1972 and is rightly regarded as one of the greatest Rangers players of the post-war era.
The club last night announced his death with 'great regret' and paid fullsome tribute to him as a club legend "in every sense" on and off the pitch. It added: "Sandy will be missed by everyone who knew him."
Ibrox manager Ally McCoist said: "There have been many great names associated with Rangers Football Club in our 142 year history and Sandy is a Rangers legend in every sense of the word.
"We are all devastated by the news he has passed away, we have lost a great man today.
"I had the privilege of watching Sandy playing for Rangers when I was a young boy, I had enjoyed the pleasure of working with him closely since I returned to the Club in 2007 and he was a truly remarkable human being.
"His achievements both on and off the pitch are second to none and I was honoured to regard him as a friend. He gave everything for this great club and worked tirelessly in a number of roles because he wanted to ensure the traditions, history and standards at Rangers were maintained.
"He was respected not only by Rangers fans but also the wider football community and he is a huge loss to the game. We will never see his like again in the modern era.
"He recently told me he was proud to be a Ranger and wanted to be remembered forever as a Ranger. Well Sandy you will go down in history as one of the greatest of all time and we will miss you terribly."
He is surivved by his wife Shona, children Steven and Nicola, and his grandchildren. Rangers said its thoughts are with them and his friends at this time.
Rangers chief executive Graham Wallace added: "Sandy Jardine epitomised everything that is good about Rangers. He was a man of principle and class and his contribution during his career both on and off the pitch was truly remarkable.
"He was a credit to the Club for decades and in my meetings with him his dignity, class and love for Rangers shone through.
"On behalf of everyone at Rangers I wish to send my sincere condolences to Sandy's family and friends at this difficult time."
Jardine, whose full name was William Pullar Jardine, who gave five decades of service to the Club first as a player and then when he returned to Ibrox in a non-playing role.
He was twice named Player of the Year in Scotland and scored 77 goals for Rangers in almost 800 appearances.
He won three League Championships with Rangers, five Scottish Cups, five League Cups and the European Cup Winners' Cup in 1972 and is seen as one of the greatest Rangers players of the post-war era.
Jardine was also praised by the club for his role following the club's descent into administration during the Craig Whyte era two years ago. i
He was a key member of the Former Rangers Players' Benevolent Fund and the driving force behind the Rangers Fans' Fighting Fund which was launched during the dark days of administration.
He led a march of thousands of Rangers fans to Hampden Park during the administration to protest at the sanctions imposed on the Club that threatened its very existence at that time.