Bony is expected to leave Vitesse this summer, with a host of clubs across Europe chasing his signature.
Swansea were thought to be front-runners for Bony after tabling a bid for the Ivory Coast international last week.
However, West Ham have now entered the race for Bony, with Sam Allardyce keen to lure him to Upton Park.
Allardyce has already brought in Andy Carroll on a permanent basis from Liverpool and now he is understood to have lodged a £10million bid for Bony.
Bony is thought to be keen on a move to England and he could have a choice over where he moves, with Swansea and West Ham bidding for his services.