There are still some details to be sorted out between Mbappe's attorney and LaLiga champions Madrid, but nothing that would prevent the deal from happening, sources added.
PSG were confident of Mbappe staying as recently as last week, sources said, and they tried everything they could to convince the 23-year-old, who was voted Ligue 1 player of the season by his peers for a third season in a row, to remain with the club after five years at the Parc des Princes.
They offered Mbappe a huge new deal, promised him a better squad for next season, offered him the captain's armband and 100% of his own image rights, according to sources.
For Mbappe, who is having the best season of his career with 25 goals and 17 assists in 34 Ligue 1 matches, it was never about money, sources said. His ambition was always to play for Real Madrid one day, and there is a feeling right now in Paris that he will fulfill his dream.
Mbappe said on Monday he is "almost" ready to announce where he will play next season as Real Madrid continue to chase his signature.
His contract at PSG expires this summer, although head coach Mauricio Pochettino said last month he is "100% sure" that Mbappe will sign a new contract at the club.