According to the Daily Mirror, Chelsea will improve their offer with Everton wanting £34m for young England international.
The Merseyside club have twice publicly announced that the 21-year-old is not for sale, but the publication believe Stones is available should the offer reach their asking price.
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They are also reporting that Manchester City are waiting in the wings should Chelsea have a bid accepted for Stones, having been long term admirers of the former Barnsley player.
Manchester United are also reported to be interested in Stones, who has four caps for England, and following their recent set back in their chase of Sergio Ramos, may step up their interest.
Manchester City are currently on high alert as to the situation surrounding Stones and are ready to pounce if Chelsea have a bid accepted. The Etihad club would prefer to see the player complete a full season at Goodison Park but may be forced in to action by their title rivals.
The blue side of Manchester are desperate to add young English talent to their side, as shown by the signings of Raheem Sterling, Fabian Delph and Patrick Roberts.
Chelsea have already had two bids of £20m and £25m rejected, with Everton boss Roberto Martinez stating his disappointment at Jose Mourinho's public pursuit of the young centre back.
If Everton's asking price was to be matched by any of the clubs in question, it would break the current British transfer record for a defender, which would break the current record set at £31.9 million by Elaqium Mangala's move to City from FC Porto.
Mourinho has publicly stated he would be prepared to break that record in order to land his top target and would be prepared to play him over 30 times in the coming season, hinting that he sees the defender as more than worthy competition to the current centre back partnership.
A tug of war for the defenders services is looking likely to take place should Chelsea take the initial step of match Everton's asking price.
The Merseyside club could yet keep hold of their prized asset for another season as they look to improve on a disappointing previous campaign.
The English defender has a big future ahead of him, but it is yet to be decided where that future lies.