Amid uncertainty over his place under new boss Enzo Maresca, the Croatia international had been linked with a move back to Italy.
Several reports indicated Serie A duo Juventus - and former club Inter Milan - were interested in an offer, if City were willing to part ways.
The 32-year-old is now into the final year of his contract at Etihad Stadium, and is yet to be offered a renewal, after only playing nine times for City last season, due to persistent Achilles and ankle injuries.
Maresca offered a lukewarm update on Kovacic's future after Saturday's friendly vs. Inter, stating: "Mateo is our player, until the day he is here I will try all my best.
"It's my duty to try to help players give their best. If he's here, I'm happy and we will continue to work with him."
Kovacic's own answer over what the next few weeks holds was far clearer than his manager's stance.
"I don't feel question marks. They can count on me and it's important for the fans to hear that from me.
"For the players, it's always on you to show your best version, because if you're playing well, the manager will put you first."