It is understood that the Portuguese international does want a move to the Premier League and he has a £40m release clause in his contract.
Sporting have held talks with West Ham over the past two weeks but will only agree to a deal to sell Carvalho if the Hammers match the buy-out fee.
Slaven Bilic's side have had an opening bid turned down for Carvalho, who was left out of the matchday squad for their Champions League play-off first-leg tie against Steaua Bucharest, which ended 0-0.
The 25-year-old is a product of the Sporting academy and has scored nine goals in 137 appearances since making his debut in 2011. He is also a Portuguese international and helped his country to win the Euro 2016, starting in the final against France.
Sporting have recently signed midfielder Rodrigo Battaglia as a possible replacement should Carvalho leave during this transfer window.
West Ham have already made five major signings during the summer window with Pablo Zabaleta, Joe Hart, Marko Arnautovic, Sead Haksabanovic and Javier Hernandez arriving at the London Stadium.