The Reds turned to Tsimikas, a regular starter for the Greek Superleague champions, after Norwich City rejected their bid for Jamal Lewis. Liverpool reportedly offered £10 million for the 22-year-old but Norwich wanted closer to £20 million.
Tsimikas, 24, is expected to provide cover for Andy Robertson, who made 44 starts across all competitions last season.
"He's fast, capable of carrying the ball down the wing, and offers plenty of attacking output," The Athletic's James Pearce and Tom Worville wrote last month.
With 25 appearances in European competition, Tsimikas has considerable experience as well. He was in the Olympiacos side that knocked Arsenal out of the Europa League in February.