Barkley has one more year left on his contract.
Manager Koeman is not prepared to risk losing the player on a free transfer when his current £65,000-a-week contract expires in just over a year's time.
"It's up to the player but we don't wait until August".
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Though Everton will be keen to keep hold of Barkley, the chance to play in the Champions League and challenge for Premier League glory next season with Spurs may be appealing.
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"I am delighted to be here - I have been here a long time now", The Mirror quote his as saying.
As reported by the Independent, Koeman said: "It is still one week".
Asked if Barkley continuing to delay his decision is a concern, Koeman replied: "No, because I like to work with players who like to stay and we are already trying to get players in for next season in the front positions".
One player who is due to sign a new contract is Kevin Mirallas, after chairman Bill Kenwright broke the news at the club's end-of-season awards ceremony on Tuesday.
"I don't know it's up to the player and he needs to make a decision".
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The club is working on a new contract for the England midfielder. "He knows what he needs to improve", said Koeman, who had no update on the striker's future.
We would pay good, good money to find out what someone like Roy Keane makes of videos such as this.
Tottenham manager Mauricio Pochettino is renowned for managing to get the best out of talented young players and the signing of Barkley would certainly tally with the club's philosophy.
Barkley may have also registered eight assists this season (which shouldn't be sniffed at), but his goal tally is down, his shots on goal tally is down, as is his pass completion and overall attack score.
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