When asked how long Barkley had, Koeman replied: "Still one week, time for him to decide".
"I am delighted to be here - I have been here a long time now", The Mirror quote his as saying.
"It's his responsibility and the responsibility also of his agent to say yes or no because that's what the club needs to know", said Koeman, whose side host Watford tonight.
"But people tend to forget that we missed many starting players during the season". We still have three very hard games left.
"We need an answer before the end of next week or we will sell the player". I like to work with players who want to stay. "Were already trying to get players who want to play in the front positions for next season".
"It is up to the player and he needs to make a decision". One is sign. If he doesn't, we need to sell the player.
Rodriguez, Morata net braces as Madrid crush Granada in La Liga
Two first-half goals each for James Rodriguez and Alvaro Morata kept Real on course for a La Liga-Champions League double. Villarreal also need a win to bolster their Champions League aspirations. "We had a good game today".
As reported by the Independent, Koeman said: "It is still one week".
Ultimately, it doesn't really feel like Barkley is any better a player now than he was 12 months ago, or even when he first broke onto the scene back in 2013.
He has struck up a good understanding with Premier League top scorer Romelu Lukaku, and his performances in 2017 in particular have caught the attention of north London rivals Arsenal and Tottenham, with the latter touted as the favourites to land him this summer.
The 34-year-old has been at the club since joining from Sheffield United for a fee of £4m in 2007 and is contracted to the Toffees until the summer of 2018.
The Belgium global, 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.