Sakho limped out of the 1-1 draw at home to West Brom last Sunday with what the London club described as a "significant injury to his left thigh".
And speaking in a media conference ahead of the away game against Manchester United this Saturday, West Ham manager Slaven Bilic said: "Diafra Sakho, we are still waiting for the results of his scan, but it will definitely not be a couple of weeks - four to eight, maybe longer.
"We are waiting for the opinion of a couple of specialists. He has a bit of history with that thigh. It's not a light one unfortunately. Of course it's a big blow. We are not bad in terms of quantity of injuries, but unfortunately they have happened to key players."
Sakho has netted five goals from 14 appearances in all competitions and Bilic is hopeful that the other members of his forward line - including Enner Valencia - can fill the void.
Valencia has played in only three games this season but is said to be nearing a return from an ankle sprain.
Bilic added: "Enner is having a good recovery so hopefully we will have him in a couple of weeks' time. It's a good chance for the guys to step in."