The England midfielder has one year remaining on his current deal and is yet to agree on new terms despite being offered an extension by the club.
Tottenham are among a number of clubs understood to be interested in the 23-year-old with manager Koeman last week telling him to sign a new deal before the end of the season or he will be sold this summer.
And with no breakthrough forthcoming it looks more certain than ever the player will leave Goodison Park in the upcoming window.
Asked on Friday whether he was confident Barkley would put pen to paper: "No. I spoke to the player, the board spoke to his agent. I don't know if he won't give an answer. I don't know.
Koeman is relaxed and despite wanting Barkley to stay confirmed the club are pondering other options.
"We will go on and we're looking for players in that position. If he stays there will be more competition," he added. "I am not worried because I like to work with players who like to play.
"If you offer a player a new contract, and a good contract, that means you want to keep the player."Reuse content