The Blues face competition from Manchester City this summer.
He remained at Goodison Park and initially continued to impress, but recent months have seen him fall out of the first team and come in for criticism for his performances.
That could have weakened Everton's resolve to keep him should they face bids once more in the summer, and both Chelsea and Manchester City feel he is ‘gettable’ according to the Mirror.
Chelsea are particularly keen to beat their rivals to his signature, and may offer one of their young stars on loan to try to sweeten any deal.
They reportedly view Stones as the long-term successor to John Terry, who is yet to sign a contract extension and looks set to depart the club in the summer.
Everton are unlikely to be keen to sell though, with them publicly having stated their desire to keep hold of their young stars for the long term.