The 18-year-old midfielder is yet to feature in a first-team league match, but made 24 appearances in the Under-19 Bundesliga for Monchengladbach last season.
A Germany Under-18 international, Kurt enjoyed a prolific campaign, netting 16 goals in helping his side to a sixth-placed finish.
"It is significant for our excellent junior work, when Bayern are so vehemently interested in one of our talents," said Monchengladbach sporting director Max Eberl.
Eberl is disappointed that Kurt decided against extending his contract at Gladbach and feels a longer stay at the club would have proved beneficial.
"We are of the opinion that this would have been the best way for him," he said.
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