Sky Sports News claim Jose Mourinho's side have bid £20million for the former Barnsley man, who is now first-choice centre-back along side Phil Jagielka.
There have been rumours about possible interest from Chelsea in recent months as they seek a long-term replacement for ageing captain John Terry.
It is no real surprise they rate Stones highly, he has developed into a fine modern defender who is just as comfortable dribbling the ball out of defence as he is diving into a last-ditch tackle. Terry himself stated last month that he was a big fan of Stones.
"I have seen a lot of games and I am a big fan of his," Terry said. "He pulls wide of the box and wants the ball and for a young defender to do that in the Premier League takes some guts.
"He is exceptional with both feet, he is a ball-playing centre half which you need to be these days.
"He is extremely confident. He looks to play from the back and that can only be good for English football that the likes of him are coming through. He is definitely one I have been impressed by."
The 21-year-old only signed a new five-year deal last summer so Everton are in a strong bargaining position - and supporters will be hoping they remain defiant in the face of any Chelsea interest.
Stones is among a talented group of young players Roberto Martinez is looking to build his team around and so losing him would leave a huge hole in his defence.
Roberto, in denying that he is making a bid for Manchester United defender Jonny Evans said that he was looking to "build a team of younger players who are proud to wear the Everton shirt." Selling Stones would go directly against the philosophy he is trying to establish at Goodison Park.
Chelsea would have to bid significantly more in order to tempt Everton into doing a deal given Stones' potential. In fact, following the Raheem Sterling deal I consider £20million an insult.
Things would change of course should Stones get itchy feet and push for move. The hope is though, as a intelligent level-headed guy, Stones will not seek to leave Goodison so early in his career given he is not guaranteed to start regularly at Chelsea. The player, currently in Singapore for the Barclays Asia Trophy with the Blues, said on Thursday during a fan event that he was happy to be at Everton
"My aim going into the season is to try to nail down a regular spot in this Everton team and hopefully that will take care of playing for England as well," said Stones.
"I want to keep learning from the likes of Phil Jagielka and keep progressing at this club."
Martinez insisted earlier this week that Everton have "balanced the books" and have no need to sell key players. This could represent a first real test of that transfer policy should Chelsea maintain their interest.