Gomes spent the season on loan at Everton and impressed in his debut campaign in the Premier League.
According to the Times, Hammers boss Manuel Pellegrini has made signing the playmaker one of his main priorities in the summer and the club are trying to act fast to secure a deal.
Everton are keen to retain the Portuguese’s services but no preferential option was included in the deal agreed between Barcelona and the Merseyside outfit before the season started.
West Ham are now in pole position to land the 25-year-old’s signature and have tabled an opening bid of £18m.