The 23-year-old has a contract until the end of next season but has given no indication he is ready to pen an extension, putting a host of Premier League clubs on alert.
The England international has become a vital cog in Koeman's team over the course of the campaign and the Dutchman claims the club are doing all they can to convince the academy product to stay.
However, Koeman says Everton cannot afford to run the risk of Barkley leaving for free when his deal expires and has warned the midfielder he will be sold if he doesn't sign a new contract.
"Ross is a boy from Everton and we need to show him the reasons to stay," said Koeman.
"But Barkley's situation is different to Romelu because next season will be the last year of his contract.
"You make decisions at the end of the season and you sign a new contract or you sell the player. There is no 'ultimatum' but that's business - and the player needs to understand this."