Barkley, 23, rejected a new contract with his boyhood club earlier this summer and has made it clear that he is seeking a "new challenge", but Toffees boss Ronald Koeman said last week that there have been no offers for the Wavertree-born attacker so far.
Tottenham have long been linked with a move for Barkley and are hoping to conclude a deal worth around £20m for the England international, according to the Liverpool Echo. Everton placed a £50m price tag on Barkley earlier this year, but it is believed their stance has softened over their vaunted academy product and they are now ready to sell for around £30m-£35m.
Tottenham's frugal approach to the prospective signing of Barkley has opened the door for other suitors to steal a march on them, with Chelsea now seen as "serious" contenders to pinch the midfielder from under their nose of their rivals.
Antonio Conte is looking to sign British players in order to boost their homegrown quota and are believed to be prioritising the potential capture of Barkley, who made his debut for Everton in 2011.
Chelsea are looking for midfield reinforcements after allowing Nemanja Matic, Ruben Loftus-Cheek and Nathaniel Chalobah to leave the club this summer and are also suggested to be monitoring Alex-Oxlade Chamberlain's contract situation at Arsenal, though they may face competition from Liverpool for the former Southampton star. Leicester City midfielder Danny Drinkwater has also been linked with a move to the capital.
Barkley has usually played in an advanced role at Everton, but Tottenham manager Mauricio Pochettino thinks the talented but inconsistent attacker can be moulded into a more orthodox central midfielder. The Argentine is believed to view Barkley as Mousa Dembele's potential successor, with doubts over the Belgium international's ability to stay fit gradually growing stronger.
Everton manager Koeman has already prepared for life without Barkley and is hoping Swansea City talisman Gylfi Sigurdsson arrives to replace the former Leeds United loanee in the coming days.
Sigurdsson, 27, has barely featured for Swansea during their pre-season friendlies this summer and has told the south Wales outfit of his intention to leave the Liberty Stadium and move to Merseyside.
Barkley is not the only homegrown talent that looks set to leave Everton in the coming weeks. Teenage attacker Kieran Dowell is set to join Championship outfit Nottingham Forest on a season-long loan, despite impressing during Everton's pre-season friendlies in July.