The 20-year-old Gotze, who joined Dortmund as an eight-year-old, is regarded as one of the country's most talented young players, has already won two Bundesliga titles and played 22 times for Germany, having also represented them at various age levels.
Losing one of their leading players to Bayern is a bitter pill for Dortmund to swallow and comes as an unwanted distraction on the eve of their Champions League semi-final first leg at home to Spanish giants Real Madrid.
"I think there could have been a worse time to announce this - four hours before the game," said a relaxed-looking Dortmund coach Jurgen Klopp.
"I can say time does not heal all wounds but it helps the healing process. The reason we are all disappointed is because he is a great player and we would like to keep him."
"Mario Gotze and his agent informed Borussia Dortmund a few days ago that the player would like to trigger his release clause and move to Bayern Munich on July 1," the club said on their website on Tuesday.
Recently crowned Bundesliga champions Bayern confirmed the move in a "media explanation", adding that they had intended to announce the signing after Dortmund had played on Wednesday.
"We confirm that Mario Gotze will play for Bayern from July 1," they said, adding that they would match Dortmund's buyout clause.
"Together with coach Jurgen Klopp, sporting director Michael Zorc asks all Borussia Dortmund fans to unconditionally support Mario Gotze in the last games of the season, especially in the Champions League semi-final against Real Madrid, just as they would any other Dortmund professional," the statement said.
Shinji Kagawa, who went from a little-known Japanese second-division player to one of the Bundesliga's most prodigious talents in two seasons with Dortmund, joined Manchester United at the end of last season.
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