The Spain international, whose contract expires in the summer, received a proposal to renew with Madrid that convinced him financially but he has decided to move on after a season in which he has started less than half of the club's LaLiga games.
Sources told ESPN that Aston Villa, Juventus and AC Milan have also shown interest in the Mallorcan in recent months but the PSG proposal is currently the one that is proving most tempting to Asensio.
The good relationship between Jorge Mendes, Asensio's agent, and Luis Campos, PSG's sporting director, has been key in the negotiations, sources added.
PSG are planning an overhaul in the summer that will include the departure of Lionel Messi and the hiring of a coach to replace Christophe Galtier.
The Parisian team also want to part ways with Neymar and are looking for forwards to accompany Kylian Mbappe in attack, sources said. As a result, Asensio has emerged as an intriguing option because at 27 years old he would arrive with the best years of his career ahead of him and as a free agent.
More than the economic aspect, Asensio understood that his role at Madrid, in terms of minutes, was not going to change next season, hence his decision to leave the Santiago Bernabeu on June 30.