The Hammers were one of the clubs interested in the services of the Belgium international his summer, but he snubbed their offer to join Chelsea instead.
Bilic said: "I said it openly, we tried to get at the beginning one of the top strikers of the moment and they opted for a bigger club and also better money of course, but at the end we bought what we wanted to. I'm happy.
"I said a few weeks ago to the chairman and (chief scout) Tony Henry that if we want to go big game fishing, there are two conditions for us to compete with the big clubs - if we go early or if we put more money in.
"Then we might have had a chance, on paper, with those great ones.
"But we didn't do that. Then it is almost impossible to get the player."
Chelsea will face West Ham at Stamford Bridge in their season opener on Monday night.