Koeman said last week that Barkley, 23, would be sold if he did not sign a new contract by the end of this weekend.
Asked on Friday if he was confident the England international would stay, Koeman said: "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."
Everton, who will finish the season seventh in the table, are at Arsenal on the final day of the Premier League season on Sunday (15:00 BST).
It could be the final game in a Toffees shirt for Barkley, who came through the club's youth academy and has one year left to run on his deal.
"We will go on and we're looking for players in that position. If he stays there will be more competition," added Dutchman Koeman.
"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."
Everton boss Ronald Koeman is not confident midfielder Ross Barkley will stay and says he is "looking for players in that position".