Kovacic, 24, has clocked up 18 appearances for Chelsea in all competitions since joining Maurizio Sarri's side on loan, but has not done anything particularly groundbreaking as of yet. He has failed to score, and managed only one assist.
Nevertheless, the report claims that Chelsea have initiated talks with Real over a permanent move for the Croatia international. Real are said to want at least £26.8million for their on-loan star.
This will come as a blow to London rivals Tottenham, after Spanish newspaper AS claim Spurs will look to hijack any move made by Chelsea to land him on a permanent basis.
It seems as though Chelsea are indeed the favourites to land the former Inter Milan star's signature, whether that be in the January window or at the end of the season.
However, Spurs are in desperate need of a new midfielder with doubts over Mousa Dembele's future. The Belgian has entered the final months of his contract at Stamford Bridge, and remains on the treatment table with an injury.
It could see the north Londoners look to match Chelsea's bid for the midfielder when January comes around, with Real perhaps likely to recall him if a permanent offer looks to be completed.