Centre-back Hummels, 27, is the third Dortmund star to join Bayern in three years after midfielder Mario Goetze left to join the Munich giants in 2013, followed by striker Robert Lewandowski in 2014.
“For the coming 2016/17 season, the Borussia Dortmund player Mats Julian Hummels will transfer to FC Bayern Munich,” said Dortmund in a terse statement.
“Both clubs have now reached an agreement in principle and the formal completion of the transfer, and the usual medical examination of the player, should be completed in the coming days.”
Reports in the German media say Bayern have paid up to 38 million euros (US$43.26m) for Hummels, who had a year left on his Dortmund contract.
Having left Bayern in 2009 for Dortmund, Hummels is returning to the Bavarian giants having won the German league title twice with Dortmund (in 2011 and 2012) and the German Cup (2012) before helping Germany win the 2014 World Cup in Brazil.