Bernabeu chiefs have become increasingly concerned at Mourinho's admission that he has 'unfinished business' in England, and his strained relationship with the Spanish media.
On his way: Carlo Ancelotti has been tipped to leave Chelsea this summer
The pressure increased on Mourinho on Monday when former Real Madrid legend Alfredo di Stefano criticised the Portuguese's tactics in Saturday's 1-1 draw against Barcelona, saying Madrid played like a mouse against a lion of a Catalan team.
Di Stefano, 84, Madrid's honorary president, said Mourinho's team showed no personality. He praised Barcelona's superiority, saying they 'treat the ball with adoration and respect'.
Sportsmail understands Real have made Chelsea boss Ancelotti one of the leading contenders to fill the possible vacancy and have even put feelers out to gauge his readiness to take on one of the biggest challenges in the game.
Lonely figure: Jose Mourinho is under increasing pressure at Real Madrid
Ancelotti seems certain to leave Stamford Bridge at the end of the season, after being left in little doubt he will have to pay with his job for falling behind in the Barclays Premier League title race and missing out on the Champions League title.
But while he is no longer wanted by Chelsea owner Roman Abramovich, he suddenly finds himself in demand on the continent as Roma also interested in his services.
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Explore more:People: Roman Abramovich, Alfredo Di Stefano, Jose Mourinho, Carlo Ancelotti Places: Madrid, Barcelona, United Kingdom