The Germany international has established himself as a key member of Jurgen Klopp’s midfield in recent years.
And the midfielder attracted heavy interest from Juventus during the summer.
But Liverpool rejected a £23million bid from the Italian giants on transfer deadline day with the hope Can will agree an extension.
But with Can’s deal yet to be renewed he could enter pre-contract negotiations with a new club in January.
Losing Can would be a huge blow to Klopp, who has come under scrutiny in recent weeks following the Reds’ poor form.
The 23-year-old admitted towards the end of last season the reason he hadn’t signed a new contract at Anfield wasn’t down to money and was happy to remain at the club.
But The Sun claim negotiations have reached an impasse because the two parties cannot agree on a release clause.
It is reported Liverpool are not prepared to include what they consider to be a low release clause.
However, they run the risk of losing the midfielder for free at the end of the season and could be forced to cash in on him when the window reopens in the New Year.
Juventus chief Beppe Marotta confirmed at the end of last month that the club will look to reignite their interest in Can.
“Liverpool are holding on tight to Emre Can,” Marotta said.
“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.”