According to Simon Jones and Jordan Seward of the MailOnline, the Blues have asked about Kovacic as new manager Maurizio Sarri is keen to improve his options in the middle of the park.
Chelsea are considering Kovacic as a different option to Arsenal midfielder Aaron Ramsey, who they are said to hold an interest in. The Blues may reportedly wait before moving for the Welshman, though, with his contract set to expire at the Emirates Stadium next summer.
"Sarri is already scanning the market for alternatives, and Kovacic is a player he likes, having watched him closely during his days in Italy with Inter Milan," continued the report. "The Croatian international has also attracted interest from Manchester City, who are considering stepping up their pursuit of him after missing out on targets Fred and Jorginho."
Still, Real are not willing to let Kovacic leave the club this summer. According to Sergio Santos Chozas of AS, the Croatia international is keen to move on, although Los Blancos consider him too important to let go.
M.A.J joked about how talented the former Inter Milan man is:
The issue for Kovacic at Madrid is he has three of the world's best central midfielders ahead of him in the pecking order.
Compatriot Luka Modric knits the team's play together majestically, Toni Kroos dictates the tempo of matches and Casemiro is a destroyer supreme in front of the back four. The trio have been crucial to Madrid's run of three successive UEFA Champions League wins in the last three campaigns.
If any of those players is unavailable, Kovacic has tended to step in, as he's versatile and intelligent enough to replace any of the trio. However, at 24, it's understandable the midfielder wants to be a key man.
There's little doubt he's good enough to do so. As noted below, Kovacic shone for Croatia on the biggest stage during the summer:
Chelsea's midfield is looking in fine shape ahead of the campaign, with the acquisition of Jorginho a key one for Sarri's style. N'Golo Kante will also head into the season after a stellar World Cup with France.
At Napoli, Sarri typically lined up with a three-man midfield, with Marek Hamsik providing a box-to-box presence. Martin Laurence of WhoScored thinks Ramsey would be ideal for that role:
It's also a job Kovacic would be astute at doing. While he can dictate from deep, the Croatian has always appeared most comfortable when playing in a more advanced position, where can make use of both his passing range and his ability to dribble past opponents.
If the Blues can convince Madrid to let Kovacic move on, then he'd be an excellent signing and would help Chelsea forge one of the most impressive midfield setups in the division.