Kovacic has spent the last two-and-a-half years with Real, but the 23-year-old has never established himself as a first-team regular under Zinedine Zidane.
Injury has restricted Kovacic's playing time this season but despite returning to action, Real are likely to listen to offers for the former Inter Milan player.
According to Don Balon, United are interested in signing Kovacic and are prepared to offer in the region of £27m for his signature.
On Friday, manager Jose Mourinho indicated that making signings during January is under consideration and with the current uncertainty surrounding Ander Herrera and Michael Carrick remaining out of the first-team picture, a central midfielder remains on his agenda.
During his time at Real, Kovacic has scored three goals in 82 appearances in all competitions.