Stones is said to have impressed Chelsea's scouting team with his performances over the past year, and they reportedly feel that he would enhance their squad considerably.
The England international appears to have a bright future and Chelsea believe that he could form part their defence for many years to come.
Jose Mourinho's side currently have one of the best back-lines in the Premier League, but it is an ageing unit and there is a recognition that they need to start planning for the future.
Stones is still 20, so he has plenty of time to develop his game, but he already appears to have the quality to play at the very highest level.
He is not the only Everton defender Chelsea are interested in, either. The newly-crowned Premier League champions are also fans of Seamus Coleman, who they have reportedly been tracking for some time.
Mourinho apparently likes Coleman's attacking intent and feels that he could offer a good option this summer, with a big-money bid a distinct possibility.