The clubs have long enjoyed a deep connection rooted in shared values, and countless legendary figures, none more so than Sir Kenny Dalglish.
'King Kenny', as he is commonly known both north and south of the border, broke the hearts of many a Celtic fan when he left for Anfield in 1977.
Unfortunately for the Parkhead club, Dalglish cannot see the current Liverpool left-back and Scotland captain Andy Robertson moving back up to Glasgow when his contract expires in Merseyside this summer.
"If he had spoken to me about it, I wouldn't have mentioned it anyway," Dalglish said. "But Celtic, how many left-backs do Celtic need? "Why would Celtic even mark somebody's card that we're going to go for Robbo? Where's he going to play? Tierney's rattled one in again last night. He's prolific now - a goalscoring left-back.
"I don't know what's going to happen to him, but I do know that he can be proud of what he did at Liverpool. And he can be proud of what he's done for Scotland. So, if he doesn't sign for Liverpool, somebody's going to benefit.
"And if they get him, they've got a good bonus there.
"Obviously, his future is dependent on what Liverpool say to him. If they want him, I'm sure he'll sign. But I hope he signs for Liverpool."
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Much has been made of Celtic's transfer business in recent windows. The club's hierarchy has been widely criticised for a lack of ambition and market know-how.
"I think the players they have are the ones they have who are going to get the results or get the defeats," Dalglish continued. "Someone who was in charge at the time when they came in brought them in. So whoever is there now he has picked up the reins from someone else.
"And maybe they would not have signed the same players as the previous people signed.
"But once you’ve made your bed you have to lie in it, don’t you? Obviously it is either a great achievement for Hearts or the Old Firm fell away and I don’t think the Old Firm teams have reached he levels where they would be happy with.
"Both have had a lot of problems which is no one else’s fault bar their own. They have enough players. But I think a bit more consistency would not have done Rangers and Celtic any harm."
Official club records show Dalglish as having scored 167 Celtic goals in 320 appearances. The top-scoring striker at the club this season is Kelechi Iheanacho, who has found the net just three times across all competitions.
"Well, they’ve not done as well for goals as what they did last year, have they? Goals are going to win you matches," Dalglish added. "But if you don’t lose any you have a good chance as well.
"The boy at right back looks alright (Julian Araujo). And the big lad up front (Tomas Cvancara), what are they going to say about him and (Youssef) Chermiti?
"They are going to compare them, aren’t they? He has scored eight, nine goals, what’s the boy at Celtic scored? The proof of the pudding is in the eating."
Celtic striker Tomas Cvancara. (Image: Mark Runnacles / Shutterstock)
Celtic moved within five points of Hearts at the top of the William Hill Premiership thanks to their 2-1 win over Aberdeen at Pittodrie. With nine league games remaining, Dalglish believes it is Hearts' title to lose.
"If Hearts win it, they will have deserved to win it," he said. "Look at the results vs the Old Firm. It’s pretty much on Hearts’ side, isn’t it? Why would they not win it?
"If they have more points than anybody else, it’s their entitlement. I would think both Rangers and Celtic would say we have had a bad lapse at some stage of the season but it does not matter, Hearts have still been beating them.
"I know Rangers won 4-2 the last time but they were beaten the first time and second time. So if they win it is a great achievement. And it will also be deserved.
"You can have all the know-how you want, it does not make any difference if you don’t have the ability. Points make prizes, someone said on the TV show, I think. Points make prizes. You can sit and surmise and speculate about this way that way. But they have the points and deserve to be there."
Hearts might deserve to win the title based on the season so far, but O'Neill and Celtic cannot be ruled out. Across his two spells in charge at Parkhead this season, the 74-year-old has a 73 per cent win ratio.
"The first time round, he did really well. It is not a surprise," Dalglish said. "He was even better when he was here the first time round (of his interim spells).
"Great experience, great education in football, working with Cloughie (Brian Clough) must have given him a lot of things. He is knowledgeable.
"He knows what it is about. He made a couple of changes in that Old Firm game there. He got them back from being on the brink to getting a draw which could easily have turned into a three pointer as well. Martin has done excellently. He never looked sprightly when he played!
"The boy, (Wilfried) Nancy, that came in. First three games. One of them was Roma. One of them was the Cup final. And the other one was Hearts. That was a hell of a start.
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"It might have been helpful if he'd have got more information for somebody else. Somebody coming in might have been able to mark his card about players' strengths and weaknesses, do's or don'ts. It never happened for the fella.
"It was a doddle (playing against O'Neill). Cloughie got a tune out of everybody, didn't he?"
Celtic and Rangers once more lock horns this Sunday. Glasgow's big two face off in a Scottish Cup quarter-final tie at Ibrox, seven days after an enthralling 2-2 draw.
"For 45 minutes in the last game, you'd have thought Celtic were out of sight," Dalglish said. "No chance of getting anything, not a corner never mind a goal.
"They get back and by the way, with a bouncier ball, they could have won. But they did brilliantly to get back and draw 2-2.
"Whether they've learned their lesson or whether there's going to be any changes for the Rangers organisation or the way they set themselves up? Because they played well in the first half against Celtic. They were all over it. This time, I don't know if the teams will be the same.
"I can't see Rangers making too many changes. Celtic, yeah, they could go to the second half team that they put out.
"I just think also, the best atmosphere you get for a Celtic vs Rangers game is when it's here at Hampden and it's 50-50.
"Now, there's a few more fans going to be in the corner at Ibrox for the cup and it gives it a great atmosphere. I just hope that on the pitch, it's rewarded for that because it's a fantastic game.
"It's a great game to play in. I think it stands out on a level or above some of the biggest derbies that you could mention."
Dalglish scored more than a few goals against Rangers during his time sporting green and white hoops.
"My first season, really in the first team, we had three games there (at Ibrox) in September," he recalled. "The one that we got drawn in the League Cup section and we played them at Ibrox, both games, because the roof had collapsed at Celtic Park.
"And then we played them in the league game, which is always September, I think. And we won every one of them. So, if we could do it, I'm sure Martin can do it."
Kenny Dalglish was promoting Premier Sports’ live and exclusive coverage of Rangers v Celtic on Sunday from 12pm on Premier Sports 1. Watch Premier Sports via your TV provider on Sky, Virgin TV, STV Player and Amazon Prime as an add-on subscription, and on the Premier Sports’ Player.