At the end of June, reports from the Evening Standard had linked Slaven Bilic's side with a move for the Italian striker - who is expected to cost around £8 million.
The club's newly-appointed head coach is keen to give Borini another chance following his disappointing time at Liverpool; a spell in which he scored just three goals in 38 matches for the Merseyside club
The 24-year-old is widely expected to leave Anfield this summer, after he has failed to make any impact in Brendan Rodgers' first-team.
The capital outfit have already move to bring in defender Angelo Ogbonna, midfielder Pedro Obiang and forward Dimitri Payet from Juventus, Sampdoria and Marseille respectively in deals that total more than £25million.
Whether Borini would take West Ham to the level desired remains to be seen, but he did impress when given regular playing time with Sunderland two seasons ago.
The Italian scored 10 goals in 40 appearances as the Black Cats achieved Premier League survival, while emerging as Capital One Cup runners-up.
Should a move materialise, the Upton Park faithful should take positives from his spell at the Stadium of Light, rather than his disappointing form since his return to Liverpool.