The 33-year-old’s contract with Bayern Munich was set to expire at the end of the season, but the club agreed to terminate the deal in order for the veteran to join AC Milan now.
“I am leaving FC Bayern with a heavy heart, but with my head held high,” said the Dutch midfielder in a statement.
“I had four and a half wonderful and successful years here and would like to express my gratitude to the club and the fans.
“FC Bayern will always have a place in my heart.”
The Bayern statement added: “The Dutch international asked for his contract to become void to transfer for sporting reasons.”