Arnautovic has reportedly handed in a transfer request after becoming disillusioned with life at the Bet365 Stadium, as reported by Sky Sports.
Stoke boss Mark Hughes has since confirmed the situation and told Arnautovic to 'appreciate what he has' at his current employers.
Stoke City's Marko Arnautovic is a prime target for West Ham
But the East Londoners need to start getting quality through the door so if they can't come to an agreement with Stoke Bilic should switch his attentions back to another reported target, Everton's Ross Barkley.
Everton have been one of the busiest clubs in the transfer window as the Dutchman completely revamps his squad and there seems little room left for the maverick England midfielder.
Spurs are reportedly prepared to bide their time on the player so could that - coupled with the latest comments from Koeman on the situation in an interview with the Liverpool Echo - clear a path for Bilic to revisit the possibility of luring the skillful star to the London Stadium.
"At the moment it is not a clear situation," Koeman told the Echo.
"He is not fit, that's the reason he didn't travel with the team. We will see when he is fit and back, but everybody knows his situation, and we will see what will happen."
Everton manager Ronald Koeman has cast further doubt on Ross Barkley's future
The former Barcelona star went on to add that he was not finished in the transfer market and expected at least two more arrivals, one of which is rumoured to be Gylfi Sigurdsson from Swansea City - another player in Barkley's midfield position.
Should the Hammers manage to lure Barkley they will have a young, hungry player with a point to prove and potentially the perfect replacement for the departed genius of Dimitri Payet.
Barkley at 23 years of age would represent a better long-term investment that 28-year-old Arnautovic.
But Bilic is known to be looking for proven Premier League quality and there is an argument that the young Evertonian is yet to truly establish himself.