Barkley's deal at Everton runs out next summer and Koeman said last week that the 23-year-old would be sold if they fail to agree a new contract.
"It is after the game, on Monday - then we will know it".
Everton manager Ronald Koeman says he is not confident Ross Barkley will be at the club next season.
"I spoke to the player, the board spoke to his agent".
"And - it is not about his decision - we will go on. I don't need to explain more because if we don't want the player to stay longer, then we don't offer him a new contract".
"Maybe an Arsenal or Tottenham will be better for him [than Manchester City]", added Dunne, who now works as a pundit.
Koeman is relaxed and despite wanting Barkley to stay confirmed the club are pondering other options.
The academy produced attacking midfielder has been offered a new contract at Goodison Park but is yet to sign it.
"We need an answer after the weekend, I don't know what will happen if he doesn't give any answer".
"No, because it's a long time that he is thinking about his future", Koeman said.
Sunday's clash against the Gunners could, therefore, be Barkley's final game in an Everton shirt and Koeman has admitted that is already considering a replacement for the player.
Harry Kane's four-goal haul at Leicester City on Thursday sent the Tottenham striker two goals ahead of Everton's Romelu Lukaku in the race for the Premier League's Golden Boot award.
"He can improve - he knows that by himself".