Barkley has just under a year left on his contract at Everton and turned down the option of an extension towards the end of last season.
The England midfielder is expected to leave his boyhood club this summer in search of a new challenge.
Chelsea have emerged as a potential destination for the 23-year-old, with Blues manager Antonio Conte desperate to strengthen his options in midfield.
But the Blues have been dealt a setback after having their opening £25m bid turned down by Everton, according to BBC Sport.
Everton are said to want at least £50m to sell Barkley, but that valuation is expected to drop as the end of the transfer window approaches.
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Chelsea are likely to return with an improved offer, while long-term admirers Tottenham are also expected to enter negotiations.
Conte is keen to add another midfielder following the departures of Nemanja Matic, Ruben Loftus-Cheek and Nathaniel Chalobah, with Leicester’s Danny Drinkwater also said to be on his wishlist.
Toffees manager Ronald Koeman recently warned clubs the midfielder will not be sold in a cut-price deal towards the end of the window.
“No, of course not,” Koeman said, when asked if Everton would allow Barkley to leave on the cheap.
“If there is a club thinking we can wait until the last day and we put some money on the table and we get Ross Barkley, sorry, you do not get Ross Barkley.”