And the Spanish are also chasing Tottenham midfielder Christian Eriksen.
The Sun says new Real boss Santiago Solari is a huge Rashford fan and is weighing up a £50m offer.
Solari's side are reportedly hunting for a new striker after losing patience with Gareth Bale and Karim Benzema.
Madrid are also plotting a £40m bid for Eriksen, who has 18 months left on his Spurs deal.
The Dane, 26, has seen Dele Alli get a pay hike to almost £150,000 a week but talks on his new deal have stalled.