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The 20-year-old Portugal midfielder will join his new team-mates in south Wales upon his return from international duty.
Swansea manager Paul Clement worked with Sanches during his six-month spell as an assistant to head coach Carlo Ancelotti at Bayern last year, before moving to the Liberty Stadium in January.
“We wanted Renato Sanches to go to a club where he could play at a high level and have a coach who counts on him,” said Bayern chairman Karl-Heinz Rummenigge.
“He needs to play continuously for a year, which he could not guarantee with us, given the quality in the squad.”
Swansea are reported to have paid Bayern 8.5 million euros ($10.1 million) as part of the deal, which does not contain an option to buy.
Sanches joined Bayern from Benfica for an initial fee of 35 million euros last season after featuring in Portugal’s run to the European Championship title.
He made 26 appearances for the German giants as they took the Bundesliga crown, but spent most of the campaign on the bench.
Rummenigge warned Sanches that he “has to put his foot on the gas” during his time at Swansea.
Bayern have clearly not given up on the midfielder, even though he has shown none of the form that helped Portugal win Euro 2016.
The Bavarian club have refused to sell him, which reportedly put off interest from Juventus, Paris Saint-Germain and Liverpool.
“They felt Swansea was a place he could be coached and developed and where he would play more regularly,” Clement told the Swansea website.
“I’d like to thank Karl and Carlo for showing their trust in the club and myself as a coach by allowing Sanches to come.
“I’m sure he will get a warm Welsh welcome next week and we wish him great success with this.”
Swansea also announced Ivorian striker Wilfried Bony’s return to the club from Manchester City following Fernando Llorente’s departure to Tottenham Hotspur.
Bony, 28, has signed a two-year contract with an option for a third year and is reported to have cost Swansea £12 million ($15.5 million, 13 million euros).
He previously spent a year and a half at the Liberty, scoring 34 goals in 70 games, before joining Manchester City in a £28 million move in January 2015.
Swansea are currently 13th in the 20-team Premier League after a win, a draw and a loss in their opening three games of the season.
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