Cooper was in the successful Dons’ side which won the European Cup Winners’ Cup in Gothenburg in 1983, along with the Scotland boss.
The 54-year-old was taken to Aberdeen Royal Infirmary in the early hours of Sunday morning after being found injured outside his home but died on Monday afternoon.
After hearing the news McLeish, in Lima for the friendly against Peru on Tuesday night, told Press Association Sport: “I’ve been in touch with all the Gothenburg lads, made some phone calls and done some text messages and everybody is devastated at the loss of Neale.
“An amazing character and such a talented kid.
“When I arrived at Aberdeen as a young apprentice, Neale was in the system, still at school and I couldn’t believe how good he was and how good he looked on the training field for a boy of around 13 or 14.
“He went on to have a great career and was integral to Aberdeen’s success.
“We have always kept in touch and anytime I was up in Aberdeen it was inevitable that we bumped into one another.
“Neale was such a popular guy that every player was a pal of his.
“You get some players that are pally with other players, some more with their pals outside the team but Neal was the best pal of everybody.”
McLeish will have to refocus on the first of Scotland’s two post-season games, the challenge match against Peru, who are going to the World Cup for the first time since 1982.
Scotland then take on Mexico at the Azteca stadium on June 2 and the hosts are also going to Russia.
“Neale would be the first to tell us to get on with it,” said McLeish.
“We have a couple of big games for a relatively inexperienced squad but one which is full of enthusiasm and I would love to see them doing something really big in these two games.”
The former Scotland defender has sensed the excitement in Peru since the squad’s arrival earlier in the week.
He said: “They are absolutely ecstatic about qualification, having not done it since 1982 and are now on the cusp of going to Russia.
“The nation is full of fervour as you would imagine and to be honest they have given the Scottish national team a terrific welcome.
“They have massive experience over our lads but it is a great learning curve for our boys”.