Sokratis, 32, only left Arsenal last week as he agreed to terminate his contract at the Emirates after two-and-a-half seasons.
Meanwhile, Liverpool remain hugely depleted at centre-back following the serious knee injuries suffered by first-choice pairing Virgil van Dijk and Joe Gomez.
The Reds’ back-up option, Joel Matip, has also missed much of the season through injury and was not risked in the FA Cup defeat by Manchester United on Sunday.
Teenager Rhys Williams started in his place and endured a challenging fourth-round tie up against the likes of Marcus Rashford and Edinson Cavani.
The 3-2 defeat reaffirmed Liverpool’s need for a centre-back, with midfielder Fabinho continuing to play at the back and Jordan Henderson asked to fill in at times.
Jurgen Klopp recently appeared to hit out at Anfield chiefs’ failure to recruit a new defender.
But they may finally have made their move as The Athletic claims Liverpool have reached out to Sokratis.
The report notes the Reds made contact with Sokratis towards the end of last week, following his departure from Arsenal.
But widespread reports claim the free agent, who last featured in August, is on the verge of returning to his homeland to sign a contract with Olympiacos.
It remains to be seen if the prospect of reuniting with Jurgen Klopp will be enough to persuade Sokratis to stay in the Premier League.
While Borussia Dortmund boss, Klopp signed Sokratis from Bundesliga rivals Werder Bremen for €9.5million (£8.5million) in 2013 and the pair had a good relationship.
Time is running out for Liverpool to bolster their defence with much-needed recruitment, as the window shuts at 11 pm on Monday, February 1.