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."
Gotze used a release clause to opt out of his contract.
"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.
German media said Bayern would be paying around 37 million euros for Gotze which, if confirmed, would be the most ever spent on a German player.
The move means that the technically gifted attacking midfielder, who can also play on the wing, could face his future club in the Champions League final at Wembley on May 25.
Bayern face Barcelona in the other semi-final, with the first leg in Munich later on Tuesday.
Dortmund, second in the Bundesliga, immediately appealed to their fans not to turn against Gotze as they chase Champions League glory.
"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.
DOMINATION FEARS
Despite boasting average home attendances of some 80,000 and recent back-to-back league titles, Dortmund have struggled to hang on to their top players.
Playmaker Nuri Sahin left after helping Dortmund to win the 2011 Bundesliga crown and the Turk has since returned to the club on loan after unhappy stints at Real Madrid and Liverpool.
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.
Dortmund are also struggling to hold on to prolific Polish striker Robert Lewandowski.
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