Asensio has been tipped to establish himself in the Madrid first team this season and after netting twice in the two-legged Supercopa de Espana final against Barcelona, his reputation has been further enhanced.
Paris Saint-Germain are currently setting new levels in the transfer market, but Asensio has claimed that he would reject any advances from the Ligue 1 outfit.
The 21-year-old told ABC: "Thanks, but no. I am in the best place, where I want to be. I will always be grateful to Real Madrid for taking a bet on me."
Earlier this week, it was reported that Arsenal were considering whether to try to tempt Asensio to London by meeting his release clause, but it appears that the Premier League side would fail with any approach.