The midfielder is coming into the last two years of his contract.
The midfielder currently earns around £60,000-a-week at Goodison Park, but the Blues are allegedly willing to increase that significantly to ensure he signs a new deal, with a potential increase of £1million a year on the cards.
That current contract has two years left to run, but if he were to fail to agree a new deal then it would put the club in a precarious position next summer.
There would likely be plenty of clubs interested in his services, too, with the Daily Mail claiming that rivals in England and Europe would be interested despite his dip in form this season.
Naturally the Blues are not interested in a sale though, and instead wish to build their team around homegrown stars such as Barkley and John Stones.
To do so, they will need to convince the former that their plans for the future are ambitious enough to interest him. Whether they can do that remains to be seen given how Everton have under-performed in the last two seasons.