BBC Sport says City are in talks with Everton over the possible signing of England defender Stones.
A deal is yet to be agreed but it is thought the two Premier League clubs are not far apart in their valuations.
City boss Pep Guardiola has identified the Yorkshireman as the kind of central defender who fits his approach to the game.
"Normally central defenders are strong in the air and aggressive," said Guardiola on Wednesday.
"But we need to have a good build-up to create easy passes in the midfield so they can create good passes for the strikers. I believe when the ball goes from the central defender to the striker as quickly as possible, it comes as quickly as possible back.
"That is why the players in that area need quality. By 31 August we will have the right squad to play how we want."