Barkley had widely been expected to leave the Blues this summer after telling Ronald Koeman he would not be signing a new contract because he wants a new challenge elsewhere.
However, an injury picked up on the eve of the new campaign has required surgery, with reports stating he was set to miss up to three months as a result.
It had been thought that would curtail his chances of leaving Goodison Park, at least until the January transfer window.
However, as previously reported, his suitors are still keen regardless and Chelsea, one of those interested in his services, are set to step up their pursuit.
According to the Telegraph, they will look to sit down with Everton on Sunday after the two sides meet in the Premier League’s early kick off at Stamford Bridge.
Antonio Conte and his team are thought to believe that Barkley could turn out to be one of the signings of the summer and are now keen to challenge rivals Tottenham for his signature.
Spurs are longstanding admirers and are willing to pay £20 million plus £8 million in add-ons, but Chelsea are ready to better that should the talks happen.
Everton wanted £50 million at the beginning of the summer but are thought to have lowered their demands considerably, especially as Barkley could potentially leave on a free transfer next summer.
They are hoping a bidding war breaks out between the London sides to push up his valuation, though it is unlikely either will bid over £25 million up right for Barkley.
If true, the talks could present Everton with the chance to discuss a potential move for want away Chelsea striker Diego Costa.
The Spaniard is currently AWOL in Brazil and does not want to return to Stamford Bridge, favoring a return to former club Atletico Madrid.
They cannot sign players until January, though, leaving Costa in limbo. Everton are in desperate need of a striker to replace Romelu Lukaku and Costa, even on a six-month loan deal, would be an ideal solution.
Indeed, Ronald Koeman has openly stated more than once that Costa would be welcome at Goodison Park. Sunday could give him an opportunity to push the deal and it’s easy to imagine he would be more than happy to sacrifice Barkley to get it done.