The German international's current deal at the Merseyside club will run down after the end of the 2017/18 season. So far, he has decided against putting pen to paper after Jurgen Klopp's side deciding against meeting the player's demands, which is to meet the insert a release clause.
Can is a transfer target for the Serie A winners since last summer. Juventus director general Giuseppe Marotta revealed his side made an approach in signing the 24-year-old, which was turned down by Liverpool.
Marotta even confirmed that his side will do "all they can" to secure Can's services for the next season. The Italian outfit are ready to offer the midfielder a five-year deal worth £85,000-a-week ($121,290 per week).
Thompson believes Can's decision to run down his contract is to "get the big payday" and claims the player will leave Anfield if Liverpool fail to tie him down to a new contract before January.
"The Emre Can situation is a blot on the copybook as I think he has already agreed a move to Juventus," Thompson told Sky Sports.
"Players are running the contracts down to get the big payday, like Sanchez did, and if it drags on beyond January then he is definitely not going to sign a new deal at Liverpool."
After their 1-0 defeat at the hands of Swansea City, Liverpool are now fourth in the league table with 47 points after 24 games. They are two points ahead of fifth-place Tottenham Hotspur. They have signed Virgil van Dijk for £75m from Southampton and Thompson believes the addition of the Dutchman and Adam Lallana's return from injury should help them finish in top four.
"I don't see that happening, though, but with Adam Lallana getting back to fitness and Virgil van Dijk already signed, Liverpool should still have enough for a top-four finish," the Anfield legend explained.
"I can't really see any other business being done as the target has to be signing quality and that is tough to find in the final week of the winter window, unless we could convince Leipzig to let us have Naby Keita early."