The France captain, 25, has told PSG he intends to leave the club with his contract expiring in June.
Mbappe has not signed a contract with Real yet, but the deal could be announced once it is no longer possible for the clubs to meet in this season’s Champions League.
The World Cup winner is PSG’s record goalscorer with 244 goals.
Mbappe wanted his future sorted before March, so on 13 February, before training, he met PSG president Nasser Al-Khelaifi and told him he was leaving and joining Real.
After reports emerged last Thursday that he would leave, Mbappe was left out of the starting line-up for Saturday’s match against Nantes, but came off the bench to score a penalty as PSG went 14 points clear at the top of Ligue 1.
He is set to sign a five-year deal with Real, earning 15m euros (£12.8m) a season, plus a 150m euro (£128m) signing-on bonus to be paid over five years, and he will keep a percentage of his image rights.
Real manager Carlo Ancelotti has already thought how he will use Mbappe in his team, with England midfielder Jude Bellingham playing a deeper role, Brazilian Vinicius Jr playing on the left and Mbappe playing left of centre.
If Croatia midfielder Luka Modric leaves this summer, then Mbappe would inherit the number 10 shirt he wears for France.
PSG are 2-0 up against Real Sociedad after the first leg of their Champions League last-16 tie, while Real lead RB Leipzig 1-0.