The Sky Blues have been eyeing the youngster and new boss Pep Guardiola is interested in signing the 22-year-old. The Toffees have placed a £50million price tag on the England international, which would make him Britain's most expensive defender if Manchester City sign the footballer.
Last summer, the centre-back attracted the interest of Chelsea, but the Blues failed to make a deal with the England international to bring him to Stamford Bridge.
This time around, Stones is trying to force through a move to Etihad Stadium, as Guardiola has also upped his interest in the former Tykes footballer.
Stones, who signed with Everton in January 2013 from Barnsley for a fee of 3 million euros, did not appear in any of England's horrifying Euro 2016 campaign.