The Dutchman has left United after failing to break into the first team.
Fosu-Mensah hopes he can get regular playing time to show his quality.
The 23-year-old said in a statement: "[This is] a step into a really interesting and very strong league in which Bayer has been one of the best teams for years.
"I'm really looking forward to working with this team and also with [coach] Peter Bosz.
"I've talked a lot with the trainer in the past few days and conversations with Simon Rolfes and Rudi Voller have convinced me there are great prospects in Leverkusen, both for me personally and for us as a team.
"It's going to be really exciting."