The 25-year-old's agent, Horacio Gaggioli, recently revealed that his client came close to leaving the Santiago Bernabeu on loan over the summer, however a temporary move failed to materialise.
Despite one report suggesting Arsenal had failed with a £34m bid for Asensio this summer, Fichajes reports that there are now three clubs, Liverpool, Juve and Milan, who are leading the race for the winger's signature.
Jurgen Klopp is believed to be keen on bolstering his attacking options and the Reds will consider making a move for Asensio in one of the upcoming transfer windows.
Juventus are also interested in the Spaniard and feel he could provide stern competition for the likes of Paulo Dybala and Federico Chiesa, while Milan - who returned to the Champions League this season - believe they can offer Asensio regular first-team football that the other interested parties cannot guarantee.
Asensio started his first league game of the campaign against Real Mallorca on Wednesday night, and scored a hat-trick for Los Blancos in their 6-1 victory on home soil.