According to sources the 22-year-old has cost the Champions League and La Liga holders more than 80 million euros ($84 million) as well as an additional 20 million euros in bonuses.
The former Bordeaux playmaker has won 10 caps since making his Les Bleus debut in September.
“Real Madrid and Monaco have agreed on the transfer of the player Aurelien Tchouameni, who will be linked to the club for the next six seasons,” they said.
“Next Tuesday, June 14, at 12:00 at Real Madrid City, the presentation ceremony for Aurelien Tchouameni as a new Real Madrid player will take place after the corresponding medical examination,” they added.
Madrid have also signed Chelsea centre-back Antonio Ruediger ahead of the new season but missed out on Paris Saint-Germain’s Kylian Mbappe last month as he decided to stay in the French capital.