The defender started just over half of Bayern's league games last season, losing his place in the starting line-up for several matches throughout the campaign after coming under fire for poor displays at the back.
He was also heavily criticised for his comments following the 3-2 loss against Dortmund in November, with the defender claiming he had blurred vision and felt sick in the build-up to the game. Hummels was later substituted in the second half.
With Niklas Sule already part of the Bundesliga champions' squad, the 30-year-old looked in for a fight for his place in the first XI next season, and Kovac says he wasn’t interested in competing for his place.
"Mats did a great job, especially in the second half of the season," the Bayern head coach said on Tuesday.
"Like [Bayern president] Uli Hoeness said, at the end of the season Mats came to us and asked what the status quo was for the new season.
"Niklas Sule did a great job, he is a starter of the [Germany] national team and with Lucas we bought a player that we want to play centre-back. That doesn't mean anything.
"Competition means the best player will play. In that case, Mats wanted to avoid that. We accept that and the result is that he plays for BVB now."
Hummels is part of a long list of incoming transfers for Dormund this summer, with the club having also brought in Julian Brandt, Thorgan Hazard and Nico Schulz.
Having been told by Germany coach Joachim Low earlier this year that he will not feature in any future plans regarding the national team, Hummels can certainly reclaim his spot as Borussia's first-choice centre-back next season.
Hummels won three Bundesliga titles and a DFB-Pokal during his second stint at Bayern Munich, having come through the ranks without ever really breaking through before making his way with Dortmund.