Hummels, who started his career at Bayern, has opted to rejoin his boyhood team one year before the end of his Dortmund contract for an undisclosed fee.
The 27-year-old German World Cup winner, who announced his desire to leave Dortmund last month, will have to play against his future employers in the German Cup final on May 21.
Portugal's Sanches, 18, is widely regarded as one of Europe's brightest prospects and has reportedly drawn interest from several English Premier League clubs, including Manchester United.
"I have not made my decision after eight-and-a-half successful years at BVB easily," Hummels said in a statement on Dortmund's website.
"I was always proud, and still am, to wear the BVB jersey and to be part of this exceptional team. Before I return to my hometown in the summer, we have a large, common goal: to bring the DFB Cup to Dortmund once again.
"For that I will give everything, just like my teammates. I wholeheartedly thank my teammates, all the staff and fans for this long and wonderful time in Black and Yellow."
Hummels won two Bundesliga titles with Dortmund, as well as one German Cup -- including the double in 2011-12 -- and came runner-up in the 2012-13 Champions League, losing out to Bayern.
And it's not the first time Bayern has raided Dortmund for its best players, having signed Mario Goetze in 2013 and Robert Lewandowski in 2014.
Meanwhile, Benfica academy graduate Sanches has enjoyed a meteoric rise since making his first team debut for the Portuguese champion at the start of the 2015-16 season.
"FC Bayern has been watching Renato Sanches for a long period," said Bayern chairman Karl-Heinz Rummenigge.
"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."
Both Bayern and Benfica confirmed the recently crowned Bundesliga champion will pay an initial up-front fee of €35 million ($40 million), with bonus payments based on the player being included in the future World Team of the Year or named FIFA World Player of the Year.
Benfica revealed the added bonuses to be €45 million ($51 million), meaning Bayern could eventually pay $91 million for Sanches' services.
His impressive performances, both domestically and in the Champions League -- where he competed against Bayern this season -- earned Sanches his first international cap for Portugal and he could yet be a surprise inclusion alongside Cristiano Ronaldo and co. at this year's European Championships.