The Portuguese club accepts he is going to leave Estádio José Alvalade this summer and are actively seeking a Premier League club for the midfielder.
According to the British television station, the club’s agents are in contact with interested parties, chiefly the Hammers and the Toffees. Arsenal tracked the 26-year-old for a number of seasons but are understood to be looking elsewhere for their midfield solutions.
West Ham attempted to sign Carvalho last year, but couldn’t reach an agreement with Sporting over the fee, as well as harbouring concerns about his fitness levels.
Last week, the Portuguese media carried stories that Carvalho cancelled his contract, but this has since been contradicted by sources close to both the player and club.
Marco Silva, newly-appointed Everton boss, previously managed Sporting and knows Carvalho well. His tenure saw the Angolan-born midfielder firmly establish himself in the Leões first-team.
Any deal is unlikely to be completed before the World Cup. Carvalho, a member of the Portugal squad which won Euro 2016, is in Russia having been included in the 23-man squad for this summer’s World Cup finals.