Can was brought to Liverpool by Brendan Rodgers in a £10m deal from Bayer Leverkusen in 2014.
The 24-year-old has had a string of serious knee issues in the past and complications came up during his medical in Paris on Thursday.
Right-back and local lad Jon Flanagan will also depart the club ending his eight-year stay with his beloved boyhood team.More news: By the numbers - Rafael Nadal 86-2 at Roland Garros
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Liverpool's Emre Can in action with Newcastle United's DeAndre Yedlin during their Premier League match in Liverpool March 3, 2018.
Speculation about a move to the Serie A club intensified in the second half of the season and Turin now seems a likely destination for the midfielder.
Meanwhile, academy graduate Jon Flanagan, who was a mainstay in the Reds 2013/14 Premier League campaign, will also leave the club having made 51 senior appearances following his debut against Manchester City in 2011.
New contracts have been offered to U23s duo George Johnston and Conor Masterson.