The Special One, 55, is thought to be on the radar of Los Blancos president Florentino Perez for a second stint at the club.
Mourinho left Real five years ago but is still much-respected by Perez, with the Portuguese boss thought to be keen on a return.
The Galaticos have struggled domestically this season, sitting fifth in La Liga, having lost FIVE of their 14 league games.
Real have found consistency hard to come by since Zinedine Zidane left in the summer, after winning his third Champions League title.
Julen Lopetegui was quickly sacked in October, with Santiago Solari promoted from the Castilla team and handed a two-and-a-half-year deal.
But if the club's fortunes in the league don't take a sudden upturn, there's no guarantee Solari will lead the side into the 2019-20 campaign.
That opens the door to Mourinho, who has previously stated his desire to stay at Man United, despite very-public rifts with players and the board.
However, with the Red Devils enduring a torrid season in the league, the ex-Chelsea boss may not see out the final 18 months of this contract.
Mourinho spent three years at Real Madrid between 2010 and 2013, winning La Liga once, to go with a Copa del Rey crown and the Supercopa de Espana.
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