The Blues midfielder produced a stunning display to help his side to a fine 4-2 victory over Leicester , and was praised afterwards by the Dutch boss.
But Koeman also made clear that with his current deal now having barely 12 months left to run, Barkley must sign a new contract…or be sold in the summer.
Asked what he does about the tricky contract situation with the home grown 23 year old, the Goodison boss said: “We offer Ross a new contract and then two possibilities.
“One he signs the contract, and if he doesn’t sign the contract then we need to sell the player – it’s simple. It’s not so difficult in my opinion.”
Koeman was full of praise for Barkley though, along with striker Romelu Lukaku, with the pair combining for a stunning second goal, and an attacking threat that eventually destroyed Leicester.
And the Dutch manager said: “We spoke a lot about Ross and Rom because they are really important, most of the time the quality of the players can be the difference between Everton and the opponent.
“Today they played really well. We know Rom is a great finisher but Ross played really well between the lines. I think he should have scored one but it is what you want to see; they played outstanding.
“But the whole team performance was outstanding.”