West Ham signed Vlasic from Russian outfit CSKA Moscow last summer, but he struggled to make a big impact at the London Stadium during the recently concluded campaign.
Now Vlasic could be on the move to Italy, with Torino closing in on signing him on loan.
According to Italian outlet Tutto Mercato, the Croatian is an express request of Torino coach Juric.
The 46-year-old coach is keen for his countryman to be added to the Torino squad and asked the Serie A side to get the deal done.
Torino are now close to finalising a loan agreement with West Ham to take Vlasic to Italy.
The attacking midfielder has a long-term deal at the London Stadium, running until the summer of 2026, but his immediate future may lie away from West Ham.
Vlasic made a total of 31 appearances across all competitions for West Ham last term and scored once, in the Premier League against Watford.