The 18-year-old midfielder, who was loaned out to Hoffenheim midway through last season, had joined Bayern in 2008.
"In the next weeks we will decide together what the near future of David will look like and what would be the best thing for both sides," said Bayern sports director Christian Nerlinger. "We are happy to have secured such a talented young player on a long-term basis."
Bayern finished third in the Bundesliga in a disappointing season, securing at least a Champions League qualifying round spot with the final to be staged in Munich next year.
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