The full-back, part of the Rangers team which won the European Cup- Winners' Cup in 1972, died on Thursday after losing his 18-month battle with cancer.
But Greig - like Jardine born in Edinburgh but a hero to the Ibrox support - hailed a man he has known for over half a century.
Greig - who was Jardine's skipper as he led the club to their only European title when they beat Dynamo Moscow in Barcelona almost 42 years ago - said: "I have known for quite a while how ill Sandy has been and how hard he has fought. Unfortunately, it was one battle he was not able to win.
"Sandy never failed to impress people when he met them. People should remember him as an excellent football player and a gentleman. Above all that, I will look back on him as an old and trusted friend.
"I first met Sandy 50 years ago and we travelled back and forward from Edinburgh to Glasgow on the train for years.
"We lived round the corner from each other and our kids grew up knowing each other. In many ways, he was like my little brother. I will never forget him."
Former Celtic defender Danny McGrain, who played alongside Jardine for Scotland at the 1974 World Cup, spoke emotionally of a man he called "Mr Perfect".
"I only heard about it this morning and was totally taken aback as I thought he was in remission," he said.
"My wife and I had a tear in our eyes. He was a great guy, a nice, nice man.
"Nobody in the Celtic side ever said anything bad about Sandy. He was a gentleman. He will be sadly missed."