The 21-year-old has been the subject of three failed bids - the last in excess of £30million - from the Stamford Bridge club as Everton have steadfastly refused to countenance a sale.
With their position unwavering, Press Association Sport understands the England centre-back has sought to bring the situation to a head by asking to leave.
Everton's resolve will now be severely tested as, with just a week remaining in the transfer window, they have to decide whether they really can afford to maintain the position of strength they established the moment Chelsea made their interest known.
Selling Stones now would not only undermine their position on the moral high ground but would also leave manager Roberto Martinez with just one senior centre-back - Phil Jagielka - and very little time to find an adequate replacement.
Martinez has become increasingly weary fielding questions about Stones and refused to even discuss what the club's response would be should the player hand in a transfer request when he was asked about it on Tuesday ahead of a Capital One Cup trip to the defender's former club Barnsley.
"There is nothing to comment on that," he said.
"Once more, it is always the same question and always the same answer.
"The only thing that matters now is the game we have in front of us
There is nothing to announce or discuss.
"I know it is your job to keep asking the same question and unfortunately it is my job in this case to always give you the same answer."
Martinez faced a similar situation in his first summer in charge when former Everton boss David Moyes made a sustained attempt to take Leighton Baines to Manchester United with him.
Baines, having spoken to Martinez and chairman Bill Kenwright, did not agitate to leave and has subsequently admitted he "sat on the fence and saw how the thing played out".
Everton resisted United's financial might and six months later the left-back signed a new contract.
Stones's situation is different in that he has now actively asked to leave but if Everton refuse to budge then Martinez will be left having to manage an unhappy player, one of his first-choice centre-backs and a cornerstone of the defence, until at least January, when the window opens again.
Source : PA
Source: PA