Last summer, Burke left Nottingham Forest to sign a long-term deal with the emerging German outfit, who will feature in this season's Champions League, but the 20-year-old made just five starts in the Bundesliga.
Burke told the club's official website: "I feel like it is a time for me to move on and play more games. I'm here, I want to impress and I want my name to be first on the team sheet."
During his previous spell in English football, Burke scored six times in 25 appearances for Forest in the Championship, as well as featuring twice during a loan period with Bradford City.
Burke joins a West Brom side who sit in third place in the Premier League table.