Sanches, strongly linked in British media with a move to Manchester United, will join the German champions in June on a five-year contract for 35 million euros ($39.82 million).
"FC Bayern has been watching Renato Sanches for a long period," Bayern chairman Karl-Heinz Rummenigge told the club website.
"We're pleased we've been able to sign him despite big-name international competition. Renato is dynamic, a good tackler and a technically skilled midfielder who'll add even more strength to our team."
Bayern deputy chairman Jan-Christian Dreesen explained the fee his club paid for Sanches, who is widely regarded as one of the most promising young players in Europe.
"The transfer fee for the current international, who was voted the Shooting Star of Portuguese Football in his home country yesterday evening, is 35 million, 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," Dreesen said.
