Sane, 23, has been linked with the Bundesliga giant after three seasons at City, where he has won two Premier League titles.
Bayern and Germany great Matthaus is certain Sane, who is contracted until 2021, will head to the Allianz Arena.
"A player who, in my eyes, will enchant the Bayern fans next year is Leroy Sane. I am sure he will move to Munich," Matthaus said. "If [Bayern president] Uli Hoeness publicly reveals that it makes an effort for the player, this lands in the vast majority of cases in Bavaria.
"And it would be a fantastic transfer, as Sane is one of the best wingers out there. Tricky, dangerous for goal, fast, usable on both wings and provided with a great future.
"[Pep] Guardiola is not really satisfied with him, the player wants to play more regularly and Bayern looks for the departure of [Arjen] Robben and [Franck] Ribery at least one man for the wing. It would be a win-win transfer for all parties."
Bayern, which won the Bundesliga and DFB-Pokal this season, is set for something of a rebuild, with Lucas Hernandez and Benjamin Pavard joining and the likes of Robben and Ribery leaving.
Matthaus expects Bayern to break its transfer record to land Sane, having paid €80million for Atletico Madrid's Lucas.
"Quality always has its price and Sane is more expensive than the previous record transfer of €80m for Lucas Hernandez," he wrote. "He may not be quite as expensive as Neymar, but €100m is likely.
"Salary, bonus payments and transfer fees together, become a new record in Bayern's history, but the player is worth it.
"If then Timo Werner also changes to Munich, one is slowly but surely prepared for a great future. Not in this, but possibly in the next year, could [Bayer Leverkusen's] Kai Havertz land in Munich and shape with Sane an era a la Robbery."