The 23-year-old England international has been linked with moves away from Goodison Park and Koeman appears to have run out of patience with the speculation.
"Either he accepts the contract or we sell the player," Koeman told reporters ahead of Friday's Premier League match at home to Watford.
"But if you need so much time then you have doubts - I like to work with players who like to stay."
"We don't wait till August - next weekend we need an answer," added Dutchman Koeman who left Barkley on the bench for last week's visit to Swansea City.
Tottenham Hotspur are one of the clubs being linked with Barkley, who has one year left on his contract.
(Reporting by Martyn Herman; Editing by Alison Williams)
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