But the expectation is that Paris Saint-Germain will knock back Los Blancos once again and say no.
Those close to the player increasingly think the situation will eventually see a December pre-contract agreement, as time runs out in the summer window for a last sensational move.
The Independent has been told that a series of intermediaries - including Jorge Mendes - were trying to aid the deal by working out an alternative forward, but no business could be done.
That has played into PSG’s stance, who still believe that Madrid going short of their valuation - €220m (£189m) - just posturing to try and secure Mbappe on a free later in the year.
The player’s contract runs out at the end of the season, but those close to him were confident of a move in this window.
Mbappe scored a double in PSG’s win at Reims as Lionel Messi made his debut for the club.
While Hugo Lloris revealed that Mbappe is fully focused on international football with France, who play Bosnia and Herzegovina in a World Cup qualifier on Wednesday.
“As you could see from the last game with Paris. He's focused, that means he respects his job, he's professional, he's focused on every day,” said the Tottenham goalkeeper.
“He's ready to perform, that's it, what's going on with his club and Real Madrid, he doesn't belong to us, he belongs to himself.
“As you can see with the first two days of the international break, he's here with a smile and to help the national team.”
Register for free to continue reading
Registration is a free and easy way to support our truly independent journalism
By registering, you will also enjoy limited access to Premium articles, exclusive newsletters, commenting, and virtual events with our leading journalists
Already have an account? sign in
Join our new commenting forum
Join thought-provoking conversations, follow other Independent readers and see their replies