The Blues are preparing for the departure of current number one Thibaut Courtois, with the Belgian seeking out a move to Real Madrid once his contract at Stamford Bridge expires next season.
According to Italian publication 'Corriere dello Sport', Chelsea's attempts at poaching Roma goalkeeper Alisson were thwarted and they have turned their attention to Donnarumma instead.
The AC Milan shotstopper is valued at €60million, and given that the Serie A outfit have been sanctioned by UEFA and excluded from European competition for two years, there is less of a pull to stay for the 19-year-old.
Chelsea may well have to fight off competition from other top European clubs though. Donnarumma's relationship with the Milan fans is a delicate one, as his agent has been seeking out a move for the youngster for several seasons.
Donnarumma has been linked with potential transfers to Ligue 1 champions Paris Saint-Germain, Premier League outfit Liverpool and fellow Serie A clubs Napoli and Juventus.