Barkley has yet to play this season after having hamstring surgery following a training injury but is now approaching match fitness.
The England global came close to joining Chelsea in the summer transfer window after the clubs agreed a £35m fee, but ultimately opted against the move.
The Blues have retained an interest in the 24-year-old, who they came close to landing in the closing stages of the summer transfer window.
Russian Daily Oil Output Edges Up in 2017 to 30-Year High
However, rising US production, which is on the verge of breaking through 10 million bpd, has tempered the bullish outlook. Oil prices were up slightly on Thursday as a pipeline outage in Britain continued to lift Brent crude prices.
"I expect there would be some interest some time in this window if there's going to be any interest from one of the big boys".
He was rewarded with a new four-year deal in July 2014 by then-manager Roberto Martinez after featuring for England at the World Cup in Brazil.
The England worldwide is believed to be in London to negotiate terms with the reigning Premier League champions, despite having turned them down last summer.
Barkley's exit will bring an end to his association with Everton, the club he joined as an 11-year-old and represented more than 150 times.