Vlasic spent last season on loan at Torino and made a positive impression in Turin, leading to the Italians wanting to keep him.
They are trying to push West Ham’s asking price down though and are holding talks with the Hammers.
The Serie A side have offered West Ham a fee of €9m plus a further €1m in bonuses, according to Italian journalist Nicolo Schira.
The proposal is not enough for West Ham, who are demanding more to let Vlasic make a permanent move to Torino.
Torino coach Ivan Juric is keen to keep hold of Vlasic and considers him a key target for this summer.
West Ham snapped Vlasic up from Russian side CSKA Moscow, but he struggled to make an impact at the London Stadium and was shipped out on loan to Torino last summer.
He turned out in 34 of Torino’s Serie A games last term, scoring five goals and providing six assists for his team-mates.
