But the Special One has instead turned his attention to his team-mate Raphael Varane.
And Spanish outlet Don Balon report United are ready to offer the France international a huge contract to get him to leave the Bernabeu.
The centre-back is said to be happy in Madrid and Mourinho has failed in previous attempts to sign the 24-year-old.
The United boss is desperate to bolster his backline even further and is convinced Varane is the best option for his squad.
That’s despite the Red Devils completing the signing of Victor Lindelof from Benfica this summer.
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Varane has become a key player for Zinedine Zidane at Real and will not be available cheap should he move to United.
Mourinho has a number of options in defence.
As well as Lindelof, Eric Bailly, Phil Jones, Chris Smalling, Daley Blind and Marcos Rojo mean United have plenty of depth available to them.
The Red Devils return to Premier League action this lunchtime at Swansea.
Mourinho will hope his side can make it two consecutive wins after their 4-0 demolition of West Ham last weekend.