The 29-year-old goalkeeper joined Bayern from Schalke in 2020 but never established himself at the club, making just four appearances and spending most of his time out on loan.
He had a spell at Monaco before joining Stuttgart in 2023, where he remained for three seasons and helped the club win the DFB Cup. Bayern had made the decision to sell him this summer, despite Nubel still having three years left on his existing contract at the Allianz Arena.
Besiktas have also announced the signing of goalkeeper Dogan Alemdar, 23, as part of the same update on their goalkeeping department. Bayern Munich won the Bundesliga title last season by 16 points from second-placed Borussia Dortmund, though Nubel made no contribution to that success given his continued exile on loan at VfB Stuttgart.
Nubel has also featured for Germany at senior level, having originally joined Bayern with the aim of becoming long-term successor to Manuel Neuer, a role that never materialised as his career stalled behind the veteran. His move to the Super Lig gives him the chance of regular first-team football after six years largely on the periphery of one of Europe's biggest clubs.
The fee is understood to be worth up to €11.5m, made up of a fixed €6.5m payment plus a further €5m in add-ons, with Nubel also having agreed personal terms that will make him one of the better-paid players in Turkish football.
Bayern sporting director Max Eberl thanked the goalkeeper for his contribution during his time at the club, saying he had "consistently developed and established himself" during loan spells at Monaco and Stuttgart, and wished him well for the next stage of his career.
Manuel Neuer remains Bayern's undisputed number one after signing a further one-year contract extension in May, with 22-year-old Jonas Urbig viewed by the club as his long-term successor.