The 24-year-old holding midfielder has long been hailed as one of the best young players in his position in European football and built further on his reputation this summer by helping Portugal to win EURO 2016.
His performances are understood to have attracted widespread interest, and reports claim that Everton and Chelsea are among the chasing pack.
The pair are already going head-to-head for Kalidou Koulibaly of Napoli, with the Toffees thought to be prepared to match any offer from their London rivals to sign the 6ft. 5” Senegalese enforcer.
And their relationship is only likely to get frostier as the battle for Carvalho intensifies.
Everton are thought to be the front-runners as Ronald Koeman searches for a marquee summer signing to kick off his Merseyside reign, but Chelsea remain keen even though Antonio Conte has snapped up a central midfielder in the shape of N’Golo Kante to bolster his ranks.