Kovacic joined the west London club on a season-long loan in August and has become a mainstay in the first team.
The Liga club bought the Croatia international from Inter Milan for around £26.7million in 2015 and will not let him go unless they receive a significant fee.
Chelsea are willing to negotiate, having seen enough from Kovacic this season to explore whether an agreement can be struck. He has started 13 games in all competitions and played consistently alongside Jorginho and N’Golo Kante in a three-man midfield.
The 24-year-old made it clear to Madrid in the summer that he wanted to leave in order to play first-team football on a more regular basis.
He has been seen as a long-term successor at the Bernabeu to fellow Croatia international Luka Modric, who won the Ballon d’Or this week. Modric turned 33 in September and, while he is clearly still playing at a very high level, Madrid must start planning for the future.
However, the manner in which Kovacic forced through the loan move has put Madrid’s desire to keep him in real doubt.
In September, Kovacic indicated that he would be prepared to stay at Chelsea long-term. “I’m really happy at Chelsea,” he said. “The city is amazing, the fans are great, the stadium, everything is nice, so I could imagine myself here. After two years of struggling in Madrid, playing and then not playing, it was difficult. You don’t have the confidence, you don’t try the things you know.
“This was my decision to go. Chelsea came, I spoke with the coach and the club. It was definitely just Chelsea for me. I wanted to come here and I am happy that it happened.”
Chelsea and Real Madrid are on reasonable terms, but the Blues will be wary of any discussions for Kovacic being used by Madrid to talk about buying Eden Hazard.
Coach Maurizio Sarri claimed on Tuesday that Chelsea are in talks with Hazard every week about a new contract.
That is understood to be a bit of an exaggeration, but more talks have been held recently and Chelsea remain optimistic that they can persuade the 27-year-old to sign an extension, worth in excess of £300,000-a-week.
Meanwhile, Chelsea take on Wolves tonight and Ruben Loftus-Cheek said: “It is important we go into the Manchester City game [on Saturday] with a win and a bit of confidence.”