According to the Evening Standard's Simon Johnson, the Croatian initially joined the Blues on a season-long loan with no option to buy, but now his return to the Santiago Bernabeu is "in real doubt." Real paid £26.7 million to sign Kovacic in 2015, so they will want a "significant fee" for the midfielder.
Kovacic, 24, has been a key player under Maurizio Sarri this season as he has played 12 times in the Premier League—totalling 710 minutes of action—more than holding his own in the centre of midfield alongside Jorginho and N'Golo Kante.
If he carries on appearing at the same rate for the rest of the 2018-19 campaign he should accumulate close to 2,000 minutes of league action, far more than he was afforded in any of his three seasons at Real where he more regularly than not appeared from the bench.
As a result, it would be no surprise were Kovacic himself eager to make a permanent move to Stamford Bridge.
There are a number of stumbling blocks standing in Chelsea's way, though.
The first is that there was no option to buy included in the original loan, a factor that baffled many back in August when the deal was struck:
The second is the catch-22 that the better Kovacic plays for Chelsea, the more likely Real are to reject any offer of signing him permanently and take him back to the Bernabeu to take up his role as Luka Modric's successor.
Finally, as noted by Johnson, there is the fear at Chelsea that any discussions around Kovacic may be used by Real as a way to open the door to signing Eden Hazard, who has long been on their radar.
Chelsea will need to tread carefully then in their attempts to make Kovacic a permanent signing, but he has shown so far this season he could well be worth the risk.