Gomes, who spent the 2018-19 campaign on loan at Everton, was subject of an opening bid of £18 million that was rejected by the Catalonian club. It's understood Barcelona's valuation of the Portuguese midfielder may be closer to the £25-million mark.
Everton are expected to compete for Gomes' signature following an impressive season for the Merseyside outfit, while Champions League runners-up Tottenham Hotspur were linked to a £30-million approach back in April.
Manuel Pellegrini is on a mission to improve the Irons' 10th-placed standing last season. The Chilean manager has already brought in goalkeeping competition for current No. 1 Lukasz Fabianski via free swoops for Roberto and David Martin but has apparently prioritized a central midfielder in the summer transfer window.
West Ham ended the season with Declan Rice and Mark Noble patrolling the middle of midfield while Jack Wilshere and Carlos Sanchez spent long periods in the treatment room. Pedro Obiang is believed to be keen on a return to Italy.
Signing a striker also seems to be high on Pellegrini's wish list, with Fulham's Aleksandar Mitrovic and West Bromwich Albion's Salomon Rondon reportedly being considered as replacements for Andy Carroll. Alaves center-back Guillermo Maripan is another name interesting Pellegrini, Steinberg claims.