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According to Italian publication Il Corriere della Sera, Donnarumma was left in tears ahead of Milan's clash with Benevento on Sunday (AEST) after being given the ultimatum of signing a new deal or sitting out next week's key clash with Juventus.
The Italy international has spent his entire senior career at San Siro, racking up close to 250 appearances in all competitions, but he is due to become a free agent in June after failing to agree fresh terms and has been linked with a move to Juve.
Maldini is not happy with the way the club's fanbase has gone about pressuring Donnarumma into signing a new deal, however, and has made the decision to curtail all discussions with the goalkeeper for the time being.
"It is important to firmly reiterate that nobody outside of Milan can decide who stays at the club and who renews," he said. "Choices regarding what happens on the pitch are down to the coach, while the club is in charge of contractual issues.
"From this moment, every single negotiation for new contract renewals is frozen until the end of the season so that the team can focus solely on the league. In the meantime, we will continue to protect our players, as we always have done."
Veteran striker Zlatan Ibrahimovic signed a 12-month extension last month, leaving Donnarumma and Hakan Calhanoglu as the two remaining high-profile Milan stars soon to be out of contract.
Milan led the way at the top of San Siro for a large part of the campaign, but they are now battling for a top-four finish after losing back-to-back matches prior to Saturday's 2-0 win against Benevento.
The Rossoneri, who are left focusing solely on Serie A after being eliminated from the Europa League at the last-16 stage by Manchester United in March, have accrued 69 points in the league this term - their most in a single campaign since 2012-2013.