Arnautovic has agreed a five-year deal, worth £26m, and will become the club's highest-paid player.
The Hammers are keen to get the deal finalised in time for Arnautovic to fly out with the rest of the squad on Monday for the pre-season tour of Germany.
If the Austria international passes a medical, he would become manager Slaven Bilic's third signing of the summer following the arrivals of goalkeeper Joe Hart and defender Pablo Zabaleta.
Arnautovic expressed a desire to join West Ham earlier this summer and handed in a transfer request in a bid to force through a move.
