The midfielder has long been linked with a move to England after earning rave reviews in Portugal.
Both Arsenal and Manchester United have expressed an interest in the Portugal international in the past, but he has never secured a move.
That could be about to change, though, as – according to A Bola – talks with Everton are now underway.
The club’s director of football, Marcel Brands, has made contact with Carvalho’s agent and a deal could be on the cards.
New Everton boss Marco Silva is keen to sign a midfielder this summer and the 26-year-old is viewed as a viable target.