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 February 13, 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 about 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 Croatian 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.
How have we got here?
Mbappe won the French title with Monaco before joining PSG as an 18-year-old in 2017, initially on loan before a 180m euro (£165.7m) move.
Overall, he has 244 goals and 93 assists in 291 games for PSG and has helped the club to five Ligue 1 title wins.
Mbappe was set to leave PSG on a free transfer at the end of the 2021-22 season but ultimately signed a two-year contract extension, with the option of a further year.
However, after Mbappe told PSG he would not agree to the additional 12 months of the deal, he was not selected for their pre-season tour to Japan last July.
He subsequently refused to meet with representatives of Saudi Pro League side Al-Hilal, who made a world record £259m offer for him.
He later returned to first-team training following what PSG said were “very constructive and positive” talks.
Mbappe turned down a move to Real when he agreed to his current deal with PSG in May 2022.
He has a one-year extension clause, so it is expected his departure will involve either a sale with a transfer fee or financial sacrifices on the player’s part.
