Vlasic joined West Ham last summer from CSKA Moscow but struggled to live up his billing at the London Stadium.
The attacking midfielder is not part of David Moyes’ plans at West Ham and they are keen to offload him in the ongoing window.
Vlasic has admirers in Serie A in the shape of Torino while their top flight rivals Napoli are also credited with interest in him.
Il Toro are in talks with the Irons over an initial loan move for the Croatian but they are to reach an agreement.
And according to Turin based daily Tuttosport, Vlasic has communicated to Moyes and the West Ham hierarchy that Torino are the only that he wants to join this summer.
Vlasic has made it clear that even if West Ham receive an offer better than the one Torino are currently proposing, he still only wants to join the Turin side.
Torino are growing in confidence that they can get a deal over the line for Vlasic, and it has been claimed that the Hammers could even sanction his exit this week.