Bayern have paid an initial £27.5million for Sanches, but possible performance-related add-ons could raise the fee to £35.5m.
Bayern deputy chairman Jan-Christian Dreesen added: "The transfer fee for the current international, who was voted the Shooting Star of Portuguese Football in his home country yesterday evening, is €35million (£27.5m), plus possible bonus payments that could become due if for example the player is named in the World Team of the Year or FIFA World Player of the Year."
Sanches has made 30 appearances for Benfica this season, scoring two goals.
BAYERN
Bayern Munich signing Renato Sanches (right) puts pen to paper on his contract
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United were keen on bringing Sanches - who was sent off in his last game for the Portuguese club on Sunday - to Old Trafford this season.
Reports suggest the Red Devils had been scouting the Lisbon-born ace in every Benfica game this season.
United had also been linked with Dortmund centre-back Mats Hummels, who will also join Bayern this summer.