The club paid tribute to 'Big Mal' on its website, describing him as "flamboyant, brilliant and larger than life".
"Malcolm will be sorely missed by everyone at the club and beyond," said the statement on City's website.
Allison joined the Blues in 1965 as assistant manager to Joe Mercer, before taking up the top job in 1972. He also had a spell back at the club from 1979 to 1980.
Club ambassador Mike Summerbee said: "Malcolm changed football by making us train like athletes, in that respect he was ahead of his time and he was a great tactician as well.
"He was also one of the lads - in effect he was the 12th player from the sidelines but he knew how to crack the whip and we respected him.
"He was a great psychologist; he knew how to handle me and how to get more out of me. He did the same for Colin Bell, Francis Lee, Neil Young and all of that great side."
City Life president and former general secretary Bernard Halford, who knew Malcolm for more than 40 years, said, "We will never see the likes of him ever again, and he did so much for the club. The signing of Tony Book was a masterstroke, but he enhanced the careers of so many other players and they worshipped him."
City, whose flags are at half mast, will pay tribute to Allison at their next Premier League home game against Arsenal next Sunday.
There will also be an appropriate commemoration to his life and work in the memorial garden at the City of Manchester Stadium, said the club.
Source: PA
Source: PA