Representatives of the Catalan giants travelled to Finland on Thursday to watch the 18-year old winger in action for Portugal at the UEFA U19 Championships.
Quina helped Portugal beat Ukraine 5-0 at the semi-final stage and the Barca officials now plan to discuss a possible transfer to the Nou Camp.
West Ham are understood to be willing to sell Quina this summer for a fee in excess of £600,000.
In Sunday’s final, Portugal U19 will take on Italy, who beat France 2-0.