According to The Guardian, the 29-year-old could be one of a number of names set to leave Goodison Park this summer as new boss Marco Silva looks to stamp his mark on the squad, with his potential availability alerting Pellegrini's side.
Bolasie has been at Everton since signing for £25m in 2016 when Ronald Koeman was in charge, but his time with the Toffees has been hampered by a cruciate ligament injury, keeping the Congolese international out for a year, after sustaining the injury in December 2016.
While his time on Merseyside hasn't gone exactly to plan, Bolasie proved during his time at Selhurst Park that he's a dangerous wide player at the Premier League level, offering good pace and dribbling ability from his preferred left hand side, provided he puts his injury troubles behind him.
Bolasie however is just one winger that the Hammers are reportedly monitoring, with the London-based side heavily linked with Lazio winger Felipe Anderson throughout the transfer window, as some reports claim they have agreed personal terms with the Brazilian.
West Ham have been one of the busier Premier League clubs so far, bringing in Ryan Fredericks, Issa Diop and Lukasz Fabianski, with more names expected to follow.