AS says the 17-year-old has caught Zinedine Zidane's eye after some impressive performances in the Championship this season.
Sessegnon can play at left-back or on the wing, and Real scouts are said to have delivered top reports after watching him throughout the campaign.
Los Blancos already have a promising player in the same position, Theo Hernandez.
But with question marks over Marcelo's long-term future, it appears Bernabeu chiefs are keen to set wheels in motion as they pursue a successor.
However, Real face stiff competition from Manchester United, who are believed to be closing in on a £25 million agreement for Sessegnon's signature.