Kovacic is currently on loan from Real Madrid and has been a mainstay in Chelsea's first team this season, featuring regularly in a three-man midfield with Jorginho and N'Golo Kante.
The 24-year-old's driving runs and intricate passing has been a regular feature in Chelsea's midfield this term.
Real Madrid had originally planned to keep Kovacic as Luke Modric's long-term successor but the former's willingness to leave to gain first-team football has left Madrid bosses in doubt about his future at the Spanish club.
And according to the Evening Standard, Chelsea have opened discussions with Real over signing Kovacic on a permanent basis.
Kovacic indicated his desire to stay at Chelsea in September.
“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."
The Croatian's move to Chelsea came as somewhat of a shock in the transfer window - only gaining traction in the final week before the season begun.
“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," Kovacic added.
The midfielder has featured 17 times this season for Maurizio Sarri, starting in 13 of those games.
Meanwhile, Chelsea have opened contract negotiations with star-man Eden hazard and are hoping to persuade the Belgian with a contract worth in excess of £300,000 per week.
Chelsea will fear that Kovacic will be used as bait in a potential deal by Madrid to have Hazard move to the Bernabeu.
Madrid are in the market for a new star player, having lost Cristiano Ronaldo during the summer to Italian giants Juventus.
Chelsea face an incredibly tough test on Saturday when they come up against league leaders Manchester City at home.
They currently sit 10 points below City and risk dropping down the table further if they lose this weekend.
Chelsea have also failed to score against the Premier League champions since April 2017 - when the Blues were making their ascent to the title that season.