The agreement will see Mbappe sign for Los Blancos as a free agent.
The 23-year-old’s contract with PSG expires June 30, but he has so far rejected every extension the Ligue 1 club has offered him.
Mbappe will earn 50 million euros gross per season at the Estadio Santiago Bernabeu, which would make him the highest-paid player in the world.
The deal between the two parties will not be made official before the Champions League last-16 tie between Real Madrid and PSG is over.
It is also understood PSG have given up on convincing Mbappe to remain at the Parc des Princes.