The PSG midfielder spent most of last season linked with rumours to Real's bitter rivals FC Barcelona and has not played for several months for the French club after disciplinary problems.
Liverpool and United have been credited with interest although Barcelona appeared to be in pole position before PSG criticised the conduct of the La Liga champions.
That put Real on alert and reports in France suggest a move is in place.
Breaking | @ggaretier reporting that PSG midfielder Adrien Rabiot is close to a provisional agreement with Real Madrid.— Get French Football News (@GFFN) April 3, 2019
Canal+ claim Los Blancos head coach Zinedine Zidane reached out to his compatriot in order to convince him to snub the Premier League in favour of his new project in his second tenure.
If the move materialises, the 24-year-old will join at the end of the current season on a free transfer.
The PSG academy graduate has won 17 titles with the club since making his first-team debut in the 2013/13 season.