Barkley has been ruled out for up to three months after picking up a hamstring injury on the eve of pre-season.
It had been thought that had scuppered his hopes of a move away from Goodison Park this summer, given he would be unable to complete a medical.
The midfielder expressed his desire to leave the club earlier this year, turning down a new contract offer and telling manager Ronald Koeman he wanted a new challenge.
Tottenham have been the most likely destination, although interest from Chelsea has also surfaced in recent weeks, with Antonio Conte keen to add more British players to his squad.
So far, the Premier League champions are the ones to have shown firm interest, with a bid for his services being rejected earlier this week.
Spurs remain keen on his services as well and are reportedly keen to see a deal done on deadline day, so long as a £30 million price tag can be agreed.
That may be enough to tempt Everton, although given Alex Oxlade-Chamberlain is set to complete a £35 million move to Liverpool, they may be keen to hold out just a little longer yet.