The Portugal worldwide has endured a hard first season at the Allianz Arena, starting just six Bundesliga matches overall after his big-money move from Benfica.
But AC Milan want him on loan for a season with a option to buy permanently for £35m.
Backed by new Chinese owners, Milan have been busy in the transfer market, signing Juventus star Leonardo Bonucci as well as Andre Silva, Lucas Biglia, Mateo Musacchio, Ricardo Rodriguez, Hakan Calhanoglu, Andrea Conti, Franck Kessie and Fabio Borini.
Hence, moving over to the San Siro would mean limited game time for Sanches, and it would not be the right move for a player already devoid of confidence and aiming to prove himself once more.
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According to Süddeutsche Zeitung, Sanches appears to be headed for Milan on loan.
A much-vaunted player, Sanches won the Golden Boy award in 2016, and starred at Euro 2016, which Portugal won.
"I can confirm that Milan have shown interest in Renato Sanches, but there is no agreement yet", Rummenigge told German paper Bild.
Sanches' chances of first-team football have taken another hit following the big-money arrival of Corentin Tolisso.
"I have the impression that AC Milan is not ready to meet our economic demands, and as long (as that continues) nothing will be done".