But Grant does not believe Tierney should have to swap flanks for Scotland as McGrain did throughout much of the 1970s.
Celtic right-back McGrain won a chunk of his 62 caps for Scotland at left-back to accommodate Rangers defender Sandy Jardine on the other side.
Tierney followed in his footsteps last year by switching from the left to the right as Gordon Strachan worked out a way of including both the Celtic player and Liverpool’s Andy Robertson.
• READ MORE - Kieran Tierney can reach 100 caps, says Malky Mackay
But Cardiff’s Callum Paterson looks set to occupy the right-back role following his return from long-term injury and both McLeish and Grant have dropped clear hints that Tierney will be deployed in a more central defensive position.
When asked whether Tierney had genuine potential to be world-class, Grant said: “Absolutely. Tierney is a top, top player and has been playing exceptionally well for Celtic for many years now, in the big, big games, and that tells you enough.
“Gordon played him at right side, the only ones I can remember doing that are Sandy Jardine and Danny McGrain, and what top players they were. Kieran is more than capable of that.
• READ MORE - Celtic boss: Scotland will get best out of Kieran Tierney at left-back
“But I think it’s very important, when you pick the national team, you are playing players predominantly in the area of the pitch where they play for the club sides. Maybe one or two have to adjust but you don’t want everybody doing that.
“Kieran knows he can play left centre-back, he knows he can play left-back comfortably, and in the modern-day game, especially at international level and European level, there’s not a hell of a lot of crosses coming in your box. It’s all about positional play, and he is outstanding at that.
“We are fortunate enough that Andy Robertson is playing at the top of his game, playing at a top-class club with a lot of pressure on him, and what an outstanding performer he has been.
“We are very, very fortunate and there will always be places for those boys in the team, never mind the squad, that’s a guarantee.”
• READ MORE - Will Kieran Tierney spend the rest of his career with Celtic?
Grant played alongside McGrain for Celtic so is well-placed to make the comparison with 20-year-old Tierney.
“Danny was my hero, and still is to this day,” the 52-year-old said. “I was very fortunate, on my debut, that Danny was playing. So I know how good Danny was, in everything he did, and his professionalism away from the pitch.
“He was the first guy I saw going to a gym. The gym at Celtic Park was in the old stand and you needed a coat on to go there, it was that cold. But Danny was up there every day working on his body, he was so strong.
“He was one of the best defenders that has ever been for Scotland. I think he was voted one of the best in the world at that time.
“He was a wonderful man and a wonderful professional, and I think Kieran deserves to be mentioned in that breath.
“He knows he has got to win those caps for Scotland to be up there with Danny but he has the potential to go that distance.”
• READ MORE - Could Celtic’s defensive woes tarnish Brendan Rodgers’ legacy?