Talks between the two clubs have progressed well in recent days with a broad agreement on the fee leading to more detailed conversations about how the payments are structured.
One of the few remaining outstanding issues centres on how much City will pay Chelsea up front. The west London club are under pressure to raise funds through player sales by the end of this month after spending in the region of £600m during the first year of Todd Boehly and Clearlake Capital's ownership of the club only to miss out on European qualification altogether.
Any outgoings completed before June 30 will fall under the first full tax year and therefore help mitigate losses to aid compliance with Uefa's Financial Fair Play rules.
Although there are still some issues to be resolved, sources have told ESPN the expectation is a deal for the 29-year-old will be completed within the next fortnight.
Kovacic is thought to have broadly agreed personal terms and the transfer will likely progress further after he finishes his international obligations with Croatia.
Their final match comes in Sunday's Nations League Final against Spain. Kovacic initially joined Chelsea on loan from Real Madrid in 2018 before signing a permanent deal in a £40m switch a year later.
He has made a total of 221 appearances, scoring six goals and winning the Champions League, Europa League, UEFA Super Cup and the Club World Cup.