The Ivory Coast striker scored a hat-trick against Brazil last week and it is understood that Real scouts had already been to see him play.
Arsenal and Liverpool are also rumoured to be paying attention to Coulibaly, who currently plays for Sienna and has been compared in style to Didier Drogba.
But it is the link to Real and Mourinho which most excites the nine-goal forward.
Speaking before his country's defeat to France in the last 16, he said: "I want to play at the top level, but I belong to Sienna.
"When I hear about Jose Mourinho being interested in me, it is like being in Disneyland when you are six-years-old - a dreamland."