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.