The 18-year-old midfielder has attracted the attention of the Catalan giants this summer, primarily because of his impressive performances on the international stage.
Quina was part of the Portugal Under-19 side which won the Under-19 European Championships this summer, with the Portuguese squad beating Italy 4-3 after extra time in the final, which was played in Finland.
A Barcelona scout watched that game and it wasn't the first time that the club had cast their eyes over him.
Sky Sports report that West Ham are now preparing to offer Quina - who came through the ranks at Chelsea - a new and improved deal in order to deter Barcelona from making a move for the youngster.
Earlier this summer, it was rumoured that the youngster could have been available for as little as £600,000. However, West Ham now seem to be keen to keep their man.
Quina has made six senior appearances for West Ham but is yet to make his Premier League debut. The Hammers see the teenager as a player with great potential, although he only has one year remaining on his contract at the London Stadium. If senior officials at West Ham get their way, that could soon be about to change.
It is not just Barcelona who are interested in Quina this summer, as he attracted interest from Borussia Dortmund earlier this year. Fellow Bundesliga clubs Hoffenheim and Werder Bremen are also said to be interested.
Meanwhile, West Ham have approached Barcelona themselves - over a potential move for midfielder Andre Gomes.