The Manchester United boss is expected to be in the market to bolster his squad in the New Year.
They have been linked with moves for Gareth Bale and Mesut Ozil most recently.
But with Marouane Fellaini soon to be out of contract and Henrikh Mkhitaryan’s future looking more and more bleak every week, Mourinho wants to strengthen his midfield options.
And he has reportedly targeted Croatia international Kovacic as the solution to his problem.
That’s according to Spanish outlet Don Balon, who claim United are confident £27m would be enough to tempt Real into selling.
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Kovacic is yet to start a La Liga match for Real this season, although that is down in some part to his injury problems.
Prior to moving to the Spanish capital in 2015, Kovacic was one of the most highly sought after midfielders in Europe.
But he has struggled to live up to the hype at the Bernabeu, with pressure mounting on Zinedine Zidane to deliver results having fallen eight points behind Barcelona in the league.
Real have been linked with a host of world-class players in recent months such as Harry Kane, Eden Hazard and Mauro Icardi.