Madrid - specifically coach Zinedine Zidane - are huge admirers of Mbappe and the teenager revealed this week that the reigning European champions have been trying to sign him since he was 14 years old.
The two clubs met at the end of last week but Monaco do not want to sell the 18-year-old, and reports in France have suggested they could demand as much as €135 million (£118m) for his signature - comfortably topping the fee Manchester United paid for Paul Pogba last summer.
Madrid will have the means to meet those demands by selling Alvaro Morata and James Rodriguez in deals that are likely to bring in close to €150m combined, but they are unlikely to want to do so.
That has not discouraged them from sitting down with Monaco and attempting to work out a resolution, though, as they see Mbappe as the best replacement for Morata and long-term successor to Karim Benzema on the market.
He could decide to stay at Monaco as they will be a Pot 1 team in the draw for the Champions League group stage and can offer him guaranteed first-team football, giving him the platform he needs to play a major role for France at next summer's World Cup.
But Mbappe is known to be a Real Madrid fan and has previously visited the club's Valdebebas training ground. Joining them would give him a chance to learn from two of his idols in Zidane and Cristiano Ronaldo.
There will also be some doubt around Monaco that they can replicate their 2016-17 season, with Bernardo Silva already having been sold to Manchester City and Benjamin Mendy likely to follow. Fabinho also looks set to depart.
Having already decided to keep Keylor Navas and Danilo and sell James, Morata, Pepe and Fabio Coentrao, Madrid hope to conclude their transfer business sooner rather than later.