Sanches, 19, victor of the 2016 Golden Boy award and voted best young player at Euro 2016, has struggled to meet expectations at Bayern since his move last summer from Benfica.
The Rossoneri have gone all out on transfers this summer, signing the likes of Hakan Calhanoglu, Mateo Musacchio, Franck Kessie, Ricardo Rodriguez, Andre Silva, Fabio Borini, and Andrea Conti.
He helped Portugal win Euro 2016 and was named the young player of the tournament.
Bayern Munich's CEO dismissed reports that the Bavarians would be entering the race to sign Alexis, having already completed the signing of Real Madrid star James Rodríguez.
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Staying at Bayern, though, would also mean a battle to make it into the first team.
Rummenigge confirmed talks are ongoing between the two sides but did concede that Milan are yet to meet their asking price.
If Sanches wants to make it at Bayern, he will have to go out on loan and prove his worth once more, and therefore, he has to assess his options carefully this summer.
"I can confirm AC Milan are interested in Renato Sanches", Rummenigge said at Munich airport, shortly before embarking on their Asian tour of China and Singapore. At least I have the impression that AC Milan are not ready to meet our demands.
The Bavarians considered making a move for him, but there have been suggestions that the German champions are anxious about the financial repercussions of the deal.