Makaay, 32, passed a medical on Thursday morning and the move was announced shortly afterwards.
"I, and Bayern Munich, would like to thank Roy Makaay for four successful years," Bayern chairman Karl-Heinz Rummenigge told the club's official website.
"He was a very important player for Bayern who wanted to go back to his homeland. We wish him and his family all the best for the future."
Eindhoven club PSV were thought to be favourites to land Makaay, but ultimately their rivals from Rotterdam won the race.
Feyenoord are also expected to announce the acquisition of Wolfsburg defender Kevin Hofland, 28, once personal terms have been agreed and the player passes a medical.
CKeer
As expected, with Klose and Toni's arrival, heads will roll. All the best Makaay!
Also, Kevin Hofland, i remember him as a promising young defender with PSV. Alas, he didn't make it big.