The former Monaco man is due to become a free agent in under 12 months' time, and PSG's attempts to tie him down to a new contract are yet to bear fruit.
Nevertheless, with the Ligue 1 giants thought to be on the brink of signing Lionel Messi on a two-year deal, Mauricio Pochettino is thought to be optimistic about his side's chances of keeping Mbappe on the books.
However, AS reports that PSG will not be able to field Messi, Neymar and Mbappe in the same team due to financial fair play regulations, and Real Madrid are preparing to swoop in.
The report claims that Los Blancos already have £100m in the transfer kitty, while the potential sales of Martin Odegaard and Luka Jovic could see them scrape enough to afford Mbappe.
The 22-year-old boasts 132 goals from 172 games in a PSG shirt and provided the assist for Mauro Icardi's winner against Troyes in Saturday's Ligue 1 opener.