The 24-year-old, who impressed against Arsenal and Wolves in the Europa League, will wear the No 21 shirt.
“We have watched Kostas for a long time and are really happy that he has joined us. It’s the perfect news before we come back together very soon,” Jurgen Klopp said.
“He is a very good footballer with an attitude to win and to compete, and I really like his mentality. It fits perfectly with the mood and the desire we have in our dressing room already.
“He has showed already at a young age he can experience football in a different country and do well – and he has been a big part of the success Olympiacos has had to win their title this season and in their European campaigns.