Jardine had fought a battle against liver cancer in the past year.
Last year doctors told hid family he might not survive the treatment.
He said at the time: "By far, cancer is the hardest thing I've ever had to deal with. They called my family in one Friday saying I might not make it but I was lucky enough to get through that."
A spokesperson for the club said: "It is with deep regret that Rangers Football Club has confirmed the death of Light Blues legend Sandy Jardine.
"Sandy had been battling bravely against cancer for the past 18 months and was a regular visitor to Ibrox until a short time ago but he sadly passed away peacefully tonight surrounded by his family.
"Rangers has lost a man of dignity, class and honour tonight and the thoughts of everyone at the club are with his wife Shona, children Steven and Nicola, his grandchildren and friends."
Rangers manager Ally McCoist called Jardine a "truly remarkable human being."
He said: "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."
Jardine played more than 400 games for Rangers, and nearly 200 games for Hearts. T
The Edinburgh-born defender also played 38 times for Scotland.