The Blues have already had three bids for the 21-year-old centre-back rejected, with the latest understood to have been worth around £30m.
However, the London club have not been deterred, with reports claiming that a £40m offer is being prepared at Stamford Bridge.
Such a sum would make Stones the most expensive defender in the history of English football, beating the £32m fee Manchester City paid for Eliaquim Mangala – only PSG duo David Luiz and Thiago Silva have commanded bigger fees.
Everton have maintained that they have no interest in selling their four-cap England international, however, such a sum may be too good to turn down.
Stones could be a long-term replacement for Chelsea hero John Terry, who was hauled off by Jose Mourinho at half-time in the weekend defeat at the Etihad Stadium.