Baggies boss Steve Clarke is trying to beat the transfer deadline to bolster his attacking options and is hoping to bring in the duo.
West Brom have yet to score in three Premier League games this season and Clarke wants help for his current strike-force at The Hawthorns.
Sunderland are willing to offload Sessegenon with the Benin international being informed he does figure in Paolo Di Canio's long-term plans.
Sessegnon did not feature in Sunderland's defeat to Crystal Palace at the weekend and Di Canio is looking to bring in a replacement for the former Paris Saint Germain man.
Anichebe has struggled to break into the Everton starting XI this season with Nikica Jelavic and Arouna Kone ahead of him in the pecking order in Roberto Martinez's plans.
West Brom have already signed Marseille midfielder Morgan Amalfitano on a season-long loan deal on deadline day, but Clarke hopes to add Sessegnon and Anichebe before the 11pm deadline.