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The Arsenal star’s contract runs out at the end of the season and he may leave the Emirates on a free transfer.
Sanchez, 28, nearly joined Manchester City in the summer but a deal wasn’t agreed.
Arsenal couldn’t sign a replacement in time and had to reject City’s £60m offer for the Chile international.
And according to Spanish transfer outlet Don Balon, Sanchez now has his heart set on a move to Real.
They claim the former Barcelona forward is even willing to take a pay cut to head to the Bernabeu.
Don Balon add that Sanchez may not be first choice at Real under Zinedine Zidane.
Real could sign the Arsenal star and use him as a rotation player to bolster their already impressive forward line.
Starsport exclusively revealed last night that the Gunners are demanding around £30m for Sanchez should they sell him in January.
However, it’s understood that Arsenal boss Arsene Wenger would prefer a swap deal involving Sanchez and PSG ace Julian Draxler.
Arsenal are back in action after the international break when the Gunners host Tottenham in the north London derby.