'Big Mal', one of the first celebrity managers in English football, joined City in 1965 to work alongside Joe Mercer.
Together they won the English league title, FA Cup, League Cup and European Cup Winners' Cup in a golden three-year spell for the Blues.
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He also had a notable period in charge of Crystal Palace, steering the south Londoners - then in the third division - to the semi-final of the FA Cup in 1975/76.
"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," former City winger Mike Summerbee said on the club website.
"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."
Allison was a defender at Charlton Athletic and West Ham United in a playing career cut short by tuberculosis and the removal of a lung in 1958.
His managerial career included spells overseas at Galatasaray in Turkey and with Sporting Lisbon, with whom he won the league title and domestic cup in 1981-82.
City said they had put their flags at half mast and would pay tribute to Allison at their next home game against Arsenal on October 24.
