The former full-back, who played 635 times for both clubs and won 38 Scotland caps, died peacefully yesterday surrounded by his family.
Rangers manager Ally McCoist said everyone at the club was devastated. He said watching Jardine play as a young boy had inspired him, adding that the club had "lost a great man".
He added: "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."
Chief executive Graham Wallace said: "Sandy Jardine epitomised everything that is good about Rangers."
The Edinburgh-born player won three League titles at Rangers, five Scottish Cups, five League Cups and the 1972 European Cup Winners' Cup before enjoying a spell at Tynecastle, narrowly missing out on the 1986 league title and winning a second Scottish player of the year award.
He is survived by his wife Shona, children Steven and Nicola, and his grandchildren.