The Arsenal ace has been tipped to quit the Emirates this season.
Sanchez, 28, is out of contract at the end of the campaign and is unlikely to pen a new deal in north London.
The Chile international is demanding a bumper pay rise to stay at Arsenal.
Sanchez nearly joined Manchester City in the summer, with the Premier League leaders tabling a £60m bid for the attacker.
And according to Spanish transfer outlet Don Balon, the Arsenal ace could end up at Real in January or next summer.
They say Real president Florentino Perez is ready to act after the La Liga champions’ slow start to the season.
Don Balon say Perez has three offers on the table of world-class stars that could head to the Bernabeu.
Sanchez, PSG star Edinson Cavani and Borussia Dortmund’s Pierre-Emerick Aubameyang are all keen to join Real.
Perez is said to be annoyed with Bale’s injuries, with the president also well aware that Ronaldo cannot continue his remarkable career forever.