It's understood the Blues are readying a fourth record-breaking bid for Stones, believed to be closer to the £40m mark.
And former Everton man Gray predicts once the London club start to up the ante, the Merseyside club will not be able resist.
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Andy Gray predicts John Stones will not be an Everton player for long
He told talkSPORT: "Chelsea see John Stones as the ideal replacement for John Terry, without a shadow of doubt, so it is a case of how much he is worth to Chelsea.
"I think Everton know that now. That is maybe why Chelsea will have to up the ante a bit but I do think sometime this month Chelsea will take John Stones and make him their player.
“I would be amazed if, at the end of August, John Stones is still at Everton."
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He continued: “It is still a work in progress for Roberto Martinez and if the bid is £35million to £40million I don’t think Everton can turn that down.
"I would be very surprised if Roberto hasn’t got something set up should the deal happen so he is ready to move when it does – and I think he will be ready.
“Everton are not at the stage now where they can look at everything and say we don’t have to sell any of our players. We do, it is as simple as that, and I think it will happen.
“It’s sad but if £30million to £40million comes in and Roberto spends it well, there is absolutely no reason why Everton can’t benefit from it.”