The Senegalese striker requested a move away from West Ham after struggling for game time last season.
Sakho will command an initial fee of £10 million, with add-ons of £6 million.
With Sakho looking extremely likely to join West Brom, this could pave the way for Saido Berahino to leave the club.
Stoke and Crystal Palace are interested in the forward but West Brom have held off selling the 22-year-old until they find a replacement.
Berahino refused to play a large part of last season as West Brom rejected a number of bids from Tottenham, something which left the striker disgruntled.
It is now thought that with Sakho's arrival, Berahino may be finally granted his wish.