Bayern have not revealed any financial details of the deal that sees Vidal arrive from last season’s Champions League finalist Juventus on a four-year contract.
Vidal has experience of playing in the Bundesliga for Bayer Leverkusen, where he scored 15 goals during four seasons before leaving for Juventus in 2011.
“I wanted to take another big step forward in my career,” the 28-year-old Vidal told the Bavarian club’s website.
“I want to keep developing as a player and win important trophies like the Champions League. I think the best opportunity to do so is with Bayern.
“I’m thrilled about playing in the Bundesliga again and obviously hope to win it this time,”
“I didn’t manage it with Leverkusen back then but with Bayern there’s a very, very good chance.”
The 28-year-old is expected to fill the gap in coach Pep Guardiola’s team left by the departure of Bastian Schweinsteiger to Manchester United.
Vidal was brought to Germany aged 20 from Chilean club Colo-Colo by Michael Reschke, who was squad director at Leverkusen.
Reschke ended his 35-year association with Leverkusen when Bayern appointed him as a director in June.
Vidal helped Juventus to four consecutive Serie A titles during his time in Italy, and the Coppa Italia last season.
He has been capped 69 times by Chile, and helped the team to a first Copa America success in July by scoring in the penalty shootout win over Argentina in the final.