The 20-year-old arrives at Oakwell having spent two years with the German champions - racking up many appearances in their reserve side.
After Tykes boss David Flitcroft had secured his services, Jennings said: "I spoke to the manager and the assistant manager and they outlined the plans they had for me and I felt Barnsley was the right club to take me further.
"My time at Bayern Munich has given me a different experience and I have learned a lot from it which I can bring back here.
"It's a great opportunity for me to play football back in England where I want to be and at a high level with a club that has great plans for this season.
"You only have to look at the second half of last season to see that Barnsley would have been in the top six based on those results so it is very exciting.
"I can't expect to turn up and play every week. I will have to work hard and do well for the team to earn the opportunity."