The Bavarians last week clinched a record fourth consecutive league title but were eliminated for the third straight year in the Champions League semi-finals under coach Pep Guardiola.
Italian Carlo Ancelotti will take over next season with Guardiola moving to Manchester City.
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).
Hummels, pending a medical later on Tuesday, will become the third Dortmund player to join Bayern since 2013 following the arrival in Munich of Mario Goetze and Robert Lewandowski and his departure is a major blow for the Ruhr valley club.
The 27-year-old German international will sign a five year-deal to 2021.
"We are delighted that Mats Hummels decided to join us," Bayern chairman Karl-Heinz Rummenigge said.
"Mats is one of the world's best central defenders and with him we will increase the quality of our team."
Rummenigge revealed that Bayern had been watching Sanches for some time.
"We are pleased we have been able to sign him despite big-name international competition," he said.
"Renato is dynamic, a good tackler and a technically skilled midfielder who'll add even more strength to our team."
The box-to-box midfielder has become a mainstay in Benfica coach Rui Vitoria's team, playing 30 games for the club after making his first-team debut in October.
Sanches caught the eye with a mature display at Bayern's Allianz Arena when the two sides met in the Champions League quarter-finals last month.
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 is 35 million euros 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.
Hummels will return to Munich almost nine years after leaving following his failure to break into the Bayern first team.
He played over 200 games for Dortmund, has 46 international caps and was in the Germany side that won the 2014 World Cup in Brazil.
Hummels helped Dortmund win the Bundesliga title in 2011 and 2012 the German Cup in 2012. He also reached the 2013 Champions League final with the Ruhr valley club.
(Reporting by Karolos Grohmann, Editing by Ed Osmond)
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