Celtic are in a period of transition as they look for a new manager, director of football and captain in the coming weeks, while also bedding in a new chief executive and facing a squad overhaul in the close season.
Dalglish, who played for Celtic between 1969 and 1977 before serving as caretaker manager at Parkhead in 2000, sees a tough job ahead for the next boss.
He says the successful candidate would do well to take a leaf out of Rangers boss Steven Gerrard's book.
"I think the most important thing is not to underestimate Celtic," Dalglish, a McDonald's ambassador, said.
"Steven has never, ever underestimated Rangers, the size of the club. He had respect for the club, he had respect for the people who supported the club, everything about it.
"If you don’t know the history of Celtic Football Club you have got problems right away. It is a demanding job and they are right to be demanding. They want to have success. Some people might say, ‘it is only one team you have to beat!’
"But you have to play them four times in the league. It is not an easy problem to solve because Rangers are miles ahead of Celtic at the moment. As they are progressing, Celtic are in transition. The transition needs to take place quickly, or as quickly as it can."
It has been reported that Celtic target Eddie Howe is keen to wait until the summer before making a decision on his future, with any delay likely to prove difficult with the Parkhead summer rebuild in mind and the prospect of a Champions League qualifier in July.
"I don’t think (principal shareholder) Dermot (Desmond) will have a quiet summer," Dalglish added.
"The work will have started already. The frustration for people is they want to know yesterday who the manager is going to be.
"But Dermot, quite rightly, will take his time, analyse who he needs, what he wants, how they go about it, and then once the manager is appointed, who they should look to sign, who should they get in, who does he want in, what finances are there. It is a big close season for Celtic."
Dalglish's formative experiences as a boy playing football on the streets of Glasgow have made him keen to shine a light on the game's current grassroots heroes, who will be recognised in the 2021 Scottish FA Grassroots Awards, presented by McDonald's.
Dalglish, helping to launch the awards nomination process, said: "Although the volunteers weren’t able to do much football, they were helping a great deal in the community with the food kitchens and other things that were going on.
"So when the grassroots awards come up, this time it’s going to be a wee bit different. The awards will look into what they did for the community during Covid.
"If there had been lockdown when we were (young) I don’t know how we would have survived it really as we were always out.
"You put your school bag in, got your piece and off you went into the street and played. That was us. And then you got shouted in for your dinner.
"We used to play in the street. One of the boys' dads used to make small aluminium pipes – electricity conduits – that made goals for us.
"Everybody wanted to play against us in the area in friendly games on a Sunday because we had goals.
"I’ve very fond memories of growing up in Glasgow and the people who were around us."
Sir Kenny Dalglish is launching nominations for the 2021 Scottish FA Grassroots Awards, presented by McDonald’s. To nominate your grassroots hero, go to www.mcdonalds.co.uk/awards