Kovacic was in the starting line-up for the defeat to Barcelona, with coach Zinedine Zidane asking him to do a man-marking job on Leo Messi. However, after the 3-0 reverse, Kovacic was isolated for criticism by the Madrid press.
Don Balon says his agents are fielding interest from across Europe ahead of the January market.
However, Roma are regarded as favourites to land Kovacic should Real choose to sell. Monchi, their sporting director, has been using his good relationship with Real to be kept up to speed on the midfielder's situation.
Should Kovacic push to leave next month, Monchi is prepared to compete with United for his signature in the coming weeks.