The Bundesliga side will pay the Reds a loan fee of around £6m as well as funding the Belgium striker’s wages in full.
Liverpool have allowed the former Lille forward leave to gain first-team football.
Origi, 22, would have had limited chances to play at Anfield this season following the arrival of Mohamed Salah.
The Belgium international is already behind Roberto Firmino and Daniel Sturridge in the No.9 role for the Reds.
While the arrival of Salah means there is also more competition on the flanks.
Earlier today Liverpool completed the signing of Alex Oxlade-Chamberlain earlier today.
Oxlade-Chamberlain has joined from Arsenal for £35m, and signed a five-year contract at Anfield.
He said: "I’m delighted to have signed for Liverpool and a big thank-you to everyone here at the FA that has made this possible, using the facilities. It has been a massive help.
“I’m not going to say too much today; I’m here with England and we’ve got two big games coming up, so I want to focus on them.”
