The Hammers have had one bid rejected for the 25-year-old, but manager Slaven Bilic is hopeful of concluding a transfer.
Carvalho was part of Portugal's triumphant Euro 2016 squad and also featured in their third-place finish at the Confederations Cup this summer.
He has scored one goal in 37 international appearances.
Bilic signed forwards Marko Arnautovic and Javier Hernandez, for £20m and £16m respectively, and goalkeeper Joe Hart on loan this summer.
However, a side containing Hart and Hernandez lost 4-0 at Manchester United in their Premier League opener on Sunday.
Sofiane Feghouli has joined Galatasaray for a reported £4m, after a season at the Hammers.
The Algerian winger scored four goals in 27 appearances, after joining from Valencia in June 2016.
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