Bayern Munich are reportedly also interested in the 29-year-old, who is entering the final year of his contract at Stamford Bridge, but the player wants to join the newly crowned Premier League champions and is now expected to do so.
Kovacic has been at Chelsea since 2018, initially on a year-long loan before joining permanently for around £45million on a five-year deal which runs out next summer.
New Chelsea manager Mauricio Pochettino is happy for Kovacic to leave as he looks to reshape a bloated squad.
Chelsea are interested in the Sporting Lisbon defensive midfielder Manuel Ugarte, while N’Golo Kante is out of contract this summer and it remains to be seen whether he will sign a contract extension.
City, meanwhile, could also lose an experienced midfielder with Ilkay Gundogan yet to sign an extension to his contract which expires at the end of June. Arsenal are keen, along with long-time admirers Barcelona.
Kovacic has made 221 appearances for the Blues, for whom he has won the Champions League, Europa League, UEFA Super Cup and FIFA Club World Cup.