The Belgium worldwide, who is the Premier League's top scorer with 24 this season, has two years left on his deal but has been linked with a move back to Chelsea and also a switch to Manchester United.
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.
As reported by the Independent, Koeman said: "It is still one week".
One player Everton could look to replace Barkley with is Wolverhampton Wanderers' Helder Costa, who has impressed in a disappointing season at Molineux.
Koeman warned at the start of April: 'We offer him a new contract and there are two possibilities. Toffee's manager Ronald Koeman has also already indicated that the Merseyside club will sell the England worldwide, should he not agree to fresh terms soon.
"The board has been working a long time to sign Ross for a longer period".
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"Because I like to work with players who like to stay".
"I think if you need [this much time] then you have doubts". "Were already trying to get players who want to play in the front positions for next season".
He has got the talent, the potential, the ability - we've seen it before - but Everton, for whatever reason, do not seem to be getting the best out of him on a regular basis.
"That's what the club needs to know because we can't wait until the last moment, it's not good for the player and it's not good for the club".
Costa has bagged 12 goals across all competitions, adding eight assists throughout the campaign, leading to him landing four awards at Wolves' end of season dinner.
"If you see the performances of Rom overall in the season, it's better than it was".