The Hammers had a bid of £15m rejected by Stoke on Monday after the Austrian forward lodged a transfer request.
Potters boss Mark Hughes has since stated that the offer was "unreasonable".
The 28-year old signed new four-year contract with City last summer.
However, Arnautovic is unpleased with the club, claiming that they promised to sell him if he wanted to leave.
The player's representatives are said to be furious at Stoke's asking price, given it is 10 times the £2m they paid Werder Bremen for his services in 2013.