The Toffees are reported to be interested in the Sporting Lisbon man but could miss out - or be forced to pay more - if AS Monaco ramp up their efforts to sign him as a replacement for Fabinho after the Brazilian's £40 million move to Anfield this week.
The Ligue 1 side were linked with the 26-year-old earlier this month but those rumours cooled, allowing Everton to reportedly make their interest known.
A recent A Bola report claims that Carvalho is keen to reunite with Marco Silva, who is expected to be announced as the Toffees new boss in the coming weeks. The pair worked together during Silva's short tenure at Sporting CP in 2014.
But now that Monaco are back in the market for a defensive midfielder, Everton will have to make a move sooner rather than later, and could be forced to pay more than Sporting's asking price of £25 million.
The Toffees can also expect to have to offer a sizeable sum for Carvalho's wages to fend off Monaco, with the French side able to offer Champions League football next season.