Goetze, 34, has won three Serie A titles with Lazio in five seasons, the last in 2016-17, and he was a key part of the Rossoneri’s success in the Champions League, scoring 11 goals in all competitions.
Lazio’s season in Serie A ended in August after a disappointing 0-0 draw against Inter Milan in the first leg of their first-leg semi-final, while his final season with the club saw him leave on a free transfer.
He had been linked with a move to Real Madrid and Milan, but he had signed a two-year contract with Lazios.
The 31-year-old has spent most of his time on the sidelines since the 2015-16 season, with him now playing in the Bundesliga.
Goetze has been a key player in the Serie A side this season, scoring four goals in nine games.
“I want to thank all the fans who have supported me for the last three years,” he said.
“This is my first season at a new club and I am ready to prove myself.
I will take my chance and prove my worth to this team.”