The defender has been heavily linked with an exit from Goodison Park this summer, with Pep Guardiola thought to be keen on the England international.
Everton are not keen to lose one of his star men though and have slapped a £50 million fee on their man, a fee they want paid straight up.
So far a deal has not been done, but the Mirror reports that Ronald Koeman met City's sporting director Txiki Begiristain to discuss a deal last week.
The two are old friends, having played alongside each other for several years at La Liga giants Barcelona.
The meeting was far from a friendly one though as they met to discuss a potential move for Stones in the coming weeks.
According to the report, Koeman is looking at ways that he can sweeten the deal that would take the young centre back to the Etihad.
One possibility is him bringing striker Wilfried Bony to Goodison Park, with the Dutchman uncertain how committed Romelu Lukaku will be to the cause.