West Ham sent the 25-year-old attacking midfielder out on loan to Torino last summer and he made 34 appearances in Serie A last season, netting five goals while assisting six.
Vlasic had an option to buy clause included in his contract, but despite being interested in the player, Torino decided against triggering it as they felt it was too high.
Torino want to negotiate down the asking price for the player and recently made a €9m offer with added €1m in bonuses to West Ham.
However, the Hammers quickly rejected the offer and they want a transfer in the region of €13.5m for the Croatian.
Vlasic, who has three more years left on his contract with West Ham, is keen on returning to Torino and he has made David Moyes aware of his intentions.
And Torino have an air of confidence about being able to re-sign Vlasic, according to Italian daily Corriere di Torino (via Tutto Mercato).
However, West Ham are determined to get as much as possible from the Vlasic deal and it remains to be seen whether they will be able to agree a deal with Torino to let the player go.
