Can, 23, has less than ten months remaining on his current deal at Liverpool.
Jurgen Klopp is keen to hand him a new long-term contract, but the club and the Germany international are yet to reach an agreement.
And Italian outlet Corriere dello Sport say the player himself has no intention of signing fresh terms at Anfield.
It is said Liverpool rejected a deadline day bid of £23million from Juventus for Can.
But he could be off in the January transfer window with the Serie A champions aware that the former Bayer Leverkusen star will be available for just £9m.
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However, Corriere dello Sport adds Can will only agree to joining the Italian giants this winter if they qualify for the knockout stages of the Champions League.
Juventus chief Beppe Moratti confirmed the club are admirers of Can and they will look to pursue him when the transfer window reopens.
“Liverpool are holding on tight to Emre Can,” Moratta said earlier this week.
“His contract expires in June. We’ll be able to negotiate with him from January.
“We won’t hide the fact we like him.
“But currently he is a Liverpool player, so we can only admire him from afar.”