The Portugal worldwide is said to have been one of Arsenal's top targets this summer, while Italian champions Juventus have also been linked with the player.
Having starred in 119 Liga games with the Lions, Carvalho is seen as someone who is ready to make the next step in European football, though his club have a huge €44 million release clause on him.
Juve are reported to have already made contact with representatives of the 24-year-old but the player is said to favour a move to England, having always been touted to join Arsene Wenger at The Emirates.
The West Ham delegation are expected to stay in Lisbon until Tuesday and they have asked the player to convince the club to accept an offer for him by next week.
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The Hammers have completed some very good business this summer already.
"The breaking news is big breaking news for West Ham".
"What a signing this could be for West Ham United, powerhouse of a centre midfielder", Hutchinson said.
Carvalho has a £39.5m release clause in his current contract.
Meanwhile, Hammers midfielder Sofiane Feghouli is said to be wanted by Turkish outfit Galatasary and midfielder Robert Snodgrass has emerged as a target of Championship club Aston Villa.