Allegri looks increasingly likely to depart the San Siro after an appalling start to the season.
Rijkaard, who coached Barcelona to Champions League success in 2006, could step straight into his shoes, reports 7Gold.
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The reputation of the Premier League means it is the most popular and one of the most watched leagues in the world.
Five English clubs will compete in the last sixteen of the Champions League for the first time ever as Liverpool, Manchester United, Manchester City, Chelsea and Tottenham face some difficult away matches.