The publication also report that two other Premier League clubs have contacted the Primera Division winners with interest in the Portuguese midfielder, but the Hammers are believed to be the keenest side in the chase, with new boss Manuel Pellegrini insistent on making a move for the player.
Barca are said to be willing to listen to offers for the player, but the Mirror claimed last year that his release clause stands at a whopping £88.5m.
The Londoners have been quite active in the market this summer and have already brought Jack Wilshere in to augment their midfield, yet the manager is understood to be after further reinforcements and Gomes has been identified as a potential recruit.
The 24-year-old moved to the Camp Nou from Valencia two summers ago but has had a subdued career playing in the Catalan side's jersey. The club are looking to offload him this summer, per reports from Spain, but it should be quite difficult now given that he could be out of action for two months.
Gomes was in action for the Spanish side against Tottenham in a pre-season friendly on Saturday night and had to go off after suffering a hamstring injury.
Initial reports claim he is likely to be sidelined for the next two months, while teammate Denis Suarez, who also hobbled off with a similar problem, should be back in two to three weeks' time.
Given this new development regarding the player's fitness, it's unclear whether West Ham are still interested. For now, though, the news coming out of Spain only points to a speculative approach.