The France striker verbally agreed to move to the Bernabeu in February and then announced in May he would leave PSG at the end of the season.
Mbappe, 25, has now signed terms with Real Madrid and will move to Spain when the La Liga transfer window opens on 1 July.
Madrid are expected to announce the deal next week and could formally present the forward at the Bernabeu before Euro 2024.
Mbappe, a World Cup winner in 2018, is PSG’s record goalscorer with 256 goals since he joined them from Monaco on an initial loan in 2017.
He has agreed a deal with Real until 2029, 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.
The Frenchman will get the chance to play alongside Luka Modric, with the Croatian midfielder set to sign a new one-year extension with the club.
Modric, 38, came on as a substitute as Madrid won their 15th European Cup title with a 2-0 win against Borussia Dortmund at Wembley on Saturday.
The former Tottenham midfielder’s current deal expires later this month, but he is expected to stay for a further 12 months.
German Toni Kroos, 34, made way for Modric with four minutes remaining on his final appearance for the club after 10 years.
Real will hold a trophy parade through the Spanish capital on Sunday night following their record-extending success.
