The 24-year-old has spent the last three weeks on the sidelines after picking up the injury on international duty, forcing him to miss four games for club and country.
Reds boss Jurgen Klopp had hoped to have Can back in time for the second leg of his side's Champions League quarter-final tie with Manchester City next week, or even for this weekend's Merseyside derby against Everton.
According to The Mirror, though, the former Bayer Leverkusen ace may have already played his final game for the club as he is expected to be out until the summer.
Can, who becomes a free agent in two months' time, is also said to be touch and go in terms of his World Cup participation.
Juventus are the frontrunners to sign the midfielder when his contract expires, while Bayern Munich and Man City are also rumoured to be interested.