According to Marca (h/t Sport Witness), the Red Devils are admirers of the Croatia midfielder, who has stated already this summer he is considering his options, and the club made a massive offer for the Los Blancos man. The club are even reported to have appointed an intermediary to help with the deal.
Kovacic, however, isn't said to be too enamoured by the prospect of the switch to Old Trafford as he is not a fan of Jose Mourinho's style of football.
Kovacic wants to leave Real Madrid, though, and he will reportedly make that point clear to the club once again this week.
According to Simon Jones and Jordan Seward of the MailOnline, Chelsea have also approached Madrid about the prospect of signing the 24-year-old, while Manchester City have also shown an interest in the former Inter Milan man.
Kovacic will head into the 2018-19 campaign on a high, as he helped Croatia make it to the final of the FIFA World Cup this summer. While he wasn't a regular starter for the team, the midfielder produced some cameos off the bench throughout the tournament.
In truth, that's been indicative of Kovacic's career so far, as he's found himself behind world-class options for club and country. Last season he made just 10 starts in La Liga for Madrid.
That's not to say he's not an excellent player, rather Kovacic finds himself behind Luka Modric, Toni Kroos and Casemiro at the Santiago Bernabeu. However, Kovacic can be a driving force in midfield when he does feature:
It's understandable that Madrid would want to keep the player because if any of that trio mentioned need to be rested, are injured or suspended, then Kovacic has tended to be the man who comes into the XI.
Kovacic is adept at sitting in front of a defence and controlling play from deeper. However, he's arguably more comfortable further up the pitch, where he can dribble forward with the ball and put his incisive passing into effect.
Still, from the player's point of view, on the brink of what should be the peak years of his career he will want to be on the pitch more often. Football writer Liam Canning thinks he would be a fine signing for United:
Central midfield doesn't seem like an area in desperate need of strengthening at Old Trafford, with Fred arriving to join the likes of Paul Pogba, Ander Herrera and Nemanja Matic.
Madrid, meanwhile, may be in a position where they need to call upon Kovacic more frequently next term. According to various reports in Italy (h/t Football Italia), Inter Milan are interested in signing Modric from the European champions in a surprise transfer.