Per Radio Montecarlo (h/t M.A. del Pozo of AS, via MailOnline's Oliver Todd), Los Blancos are prepared to smash the world-record transfer fee to sign the 18-year-old, having already approached Monaco with bids of €100 million and €115 million.
The most ever paid for a player was the £89 million Manchester United spent on Paul Pogba last summer.
Real's upcoming bid for Mbappe would blow that out of the water and likely raise eyebrows in some quarters given the youngster has enjoyed just one fine season in his career.
He had an impressive 2016-17 campaign, netting goals across all competitions as Monaco won Ligue 1 and made the semi-finals of the UEFA Champions League, per football analyst David O'Brien:
However, given he has yet to prove himself on a regular basis at the top level, it is arguable whether he is worth the fees being reported.
According to Radio Montecarlo (via Del Pozo's report), Real's deal for Mbappe would see him remain at Monaco next season because he could not yet command a starting role in Los Blancos' first XI and would be better served continuing to play regularly in Ligue 1.
The Gunners are prepared to offer a similarly huge fee to Real's, per L'Equipe (via Get French Football News):
This would be a surprise given Arsenal manager Arsene Wenger has historically been relatively frugal in the transfer market.
Real would likely still have the edge, though. Not only are they La Liga and Champions League holders and could offer Mbappe almost a guarantee of winning major silverware, but Monaco would also likely jump at the chance to keep their star forward for another season while picking up a record transfer fee.