The two clubs have agreed an initial season-long loan deal, with Torino holding an option to make the move permanent next summer for around €15million.
Vlasic arrived from CSKA Moscow in a £25.7m deal last summer but has been a major disappointment at the London Stadium and the Hammers are ready to cut their losses.
The Croatian international made 31 appearances for the Irons last term but the vast majority came either from the substitutes’ bench or in the early stages of cup competitions.
The 24-year-old started just six Premier League matches, scoring once, in a 4-1 victory over Watford, and would have seen his playing time further restricted by the arrival of Ivory Coast winger Maxwel Cornet from Burnley.
Cornet was not registered in time to face Manchester City in the Hammers’ season-opener against Manchester City last weekend, but should be in contention for this weekend’s trip to Nottingham Forest.
Elsewhere, it has been confirmed that West Ham will face Viborg FF for a place in the Europa Conference League group stage, after the Danish side completed a 5-1 aggregate victory over B36 Torshavn of the Faroe Islands on Tuesday.