Kovacic has made more than 100 appearances for Real since joining from Inter Milan in 2015 but the majority of those have come from the bench.
The 24-year-old has been unable to cement a regular place in Real Madrids first-team and has admitted he is keen to leave the Spanish heavyweights this summer.
A number of clubs have been linked with a move for Kovacic and Premier League giants United have joined the race to secure his signature, according to the Daily Mail.
Jose Mourinho signed Brazilian star Fred earlier in the month but will look to bring another top class midfielder to Old Trafford, particularly if Marouane Fellaini leaves.
But United will find it difficult to secure a deal with Real Madrid as they value Kovacic highly and he has a £265million release clause.
Kovacic, who is currently with the Croatia team at the World Cup in Russia, has also emerged as a target for Premier League champions Manchester City.
This week, Fellaini confirmed a decision over his future at Manchester United would be made soon – he is expected to leave after turning down multiple offers to sign a new deal.
Anthony Martial, the French forward, could follow Fellaini out of the exit door after his agent revealed he wanted to leave to revive his career.
Juventus have identified Martial as a top target but United have no plans to sell the 22-year-old unless a hugely tempting offer is put on the table.