The LaLiga giants made a late move to try and bring the Frenchman to Spain but saw an opening bid of £137m turned down. The Los Blancos then returned with an improved bid of £145m plus £9m in add-ons but that was also turned down.
According to French outlet L'Equipe, Madrid's offer to sign Mbappe expired at 5pm UK time. Additional reports in Spain added that Carlo Ancelotti's side were considering increasing their offer to £172m but L'Equipe say PSG have denied this claim.
The report adds that Madrid will not increase their bid and have decided to walk away from any potential deal for the 22-year-old. PSG are desperate to hold on to Mbappe, who is out of contract next summer.
The club's transfer chief Leonardo was furious with Madrid's attempt to sign Mbappe and accused them of an 'illegal' approach. He said: "PSG's position has always been to keep Kylian. That's been the aim for the last two years and it remains unchanged.
"We made him (Mbappe) a big offer two months ago – at the level of the best players – and another even more recently which was even higher, but this is a strategy of Real Madrid, to have the name of one of our players.
"For the last two years the behavior of Madrid has been incorrect, illegal, getting in contact with his representatives, unacceptable, completely incorrect towards us. And the proof of this strategy is to present an offer seven days before the market shuts and a year before the end of his contract. It seems like a strategy that suggests they have tried everything.
"It's not the sum [offered] which we believe to be a long way from what we were thinking. And we are not going to sell a player for less than we paid for him when he was 18. We have to give a certain amount to Monaco and the offer is not sufficient."