The midfielder spent last season on loan at Torino and he has been clear about wanting to return to the club in the ongoing transfer window.
The Italian club allowed the €13m option to buy to lapse with the hope of signing him for a lower fee but West Ham rejected their initial €9m offer.
Vlasic has been outspoken about his frustrations at the pace of the negotiations and even had heated talks with Hammers boss David Moyes and new director of football Tim Steidten.
Torino tabled a new bid worth €10m earlier this week and according to La Stampa (via Tutto Mercato), a new meeting is scheduled between the two clubs.
It is unclear whether the new offer has been accepted but there is a willingness from all parties to conclude a deal soon.
West Ham and Torino are set to hold a summit at the end of this week or early next week to discuss Vlasic.
The two clubs are expected to try and close out the deal for the Croatian to join Torino on a permanent deal.
