Los Blancos are long-time admirers of the Borussia Dortmund winger, according to the Daily Mail.
Sancho has scored four goals in 11 appearances this season and has established himself as one of the brightest young talents in Europe since his breakthrough last year.
Madrid were interested in signing him when he left Manchester City as an academy player in 2017, but the England international decided to move to the Westfalenstadion in search of senior football.
Now the Spanish giants are reportedly laying the groundwork for a transfer, with Dortmund resigned to losing the 19-year-old. Manchester United and PSG are also interested, though Dortmund want a transfer fee in excess of £100m.