AC Milan was outwitted in the final by Juventus who mauled the Rossoneri 4-0 with Donnarumma culpable for two of the goals.
Gattuso told Rai Sport: "At this moment Donnarumma is the Milan goalkeeper, the rest is down to the club".
"It will be a hard game, and it will take a great Milan to win".
"The lads played an extraordinary game and deserved this win, taking nothing away from Milan who had a good first half", Juve coach Massimiliano Allegri said.
Seen as the natural successor to legendary Italian 'keeper Gianluigi Buffon he's earned five worldwide caps to date keeping two clean sheets in the process.
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Kalinic, who has not reached expectations at Milan, then added to the misfortune by finding his own net with his back to the goal by the near post as he attempted to defend a corner.
Kalinic scored an own goal after 76 minutes.
Juve, meanwhile, struggled to carve out clear-cut chances in the first half and were restricted to efforts from distance, none of which were troubling for Gianluigi Donnarumma.
"Today I feel this incredible pressure on me, as if the fans were on my shoulders", said the 40-year-old. "I felt bad for a few days, but fortunately the lads completed the job and I thank them for being close to me".
The Bianconeri can clinch a record-extending seventh consecutive Serie A title with only a draw against Roma on Sunday.
"Every year Juventus manage to build up a very competitive group of players with the aim of keep winning".