The 48-year-old Juventus manager has done exceptionally well in Serie A and will join the Premier League champions next season.
It is understood that Roman Abramovich was keen on signing the Bayern Munich manager Pep Guardiola, but the Spaniard has already agreed to join Manchester City.
Allegri won the Serie A with Juventus last season and guided them to the Champions League final. The Italian manager is keen to experience the Premier League and has agreed to join the Blues according to reports from Italy.
According to the report, the Premier League champions have moved swiftly to secure the services of the Juventus manager by offering him a four-year contract worth around £21.