European champions Real Madrid have been heavily linked with a move for Courtois, whose current contract at Chelsea expires in 2019.
Talks over an extension have yielded little success so far, raising speculation that the Blues will be forced to cash in on the 25-year-old this summer.
Chelsea remain keen to keep Courtois at Stamford Bridge, but Corriere dello Sport reports that they have lined up Donnarumma as a potential transfer target should they fail to agree terms with their current number one.
Donnarumma was eager to leave Milan last summer and is again expected to push for an exit when the transfer window reopens following a breakdown in the relationship between his agent Mino Raiola and the club's hierarchy.
The 19-year-old Italy international has already made more than 100 first-team appearances for Milan since breaking into the first team and is considered one of the brightest young goalkeeping talents in world football.
However, Chelsea could face competition for his signature from Paris Saint-Germain, who are also in the market for a new shot-stopper.
Milan, meanwhile, are understood to be closing in on the signing of former Liverpool keeper Pepe Reina from Napoli as a replacement for Donnarumma.