The 26-year-old has previously been linked with a move to the Premier League, but it never materialised. However, the latest report claims that the Portugal international will leave the club in the summer transfer window, and that he will join the Toffees.
Everton have appointed Marco Silva as their new permanent manager, and he is keen to make Carvalho his first signing at the Merseyside club.
Carvalho is a highly rated midfielder, and was one of the most-wanted footballers in Europe when he helped Portugal lift the European Championship in 2016.
According to a recent report from A Bola, Silva is a big fan of Carvalho, and knows him well from his short stint with Sporting.
Silva was the manager of the Lisbon giants during the 2014-15 season, and Carvalho played a significant role, helping Sporting win the Taca de Portugal trophy at the end of the campaign.