Zinedine Zidane, Mbappe’s fellow Frenchman, is said to have convinced the Manchester United and City target to join him at the Santiago Bernabeu.
Club president Florentino Perez believes Mbappe, who has scored 22 goals in 36 games this season, fits the profile to become one of the club’s Galacticos.
The summer transfer will cost Real Madrid around £85 million if they want to convince Monaco into cashing in on their latest wonderkid.
Mbappe was initially reluctant to leave Stade Louis II as he wants to continue to play on a regular basis but Zidane appears to have landed his No 1 target.
If Mbappe goes to Madrid, that will set up a domino effect as his arrival will mean the departure of Karim Benzema
Manchester City had a £60m offer for Mbappe turned down by Monaco last summer and Pep Guardiola will be frustrated at failing to push through the move as Mbappe has gone from strength to strength during the last few months.
Mbappe has proven his class in the Champions League this season, scoring five goals and helping lead Monaco to the Champions League semi-finals, where they will be hoping to overturn their 2-0 deficit against Juventus next week.