Barkley flourished at the start of last season, but his performances towards the latter end of the campaign dropped off slightly.
Roberto Martinez always defended the England international, but Koeman has already issued a challenge to the 22-year-old.
“He needs to improve,” the Dutch boss said. “He is not a talent anymore, he is a national team player and the expectations of him are high.”
Barkley has played in a range of positions for Everton, with the midfielder being handed more defensive responsibility by Martinez last season.
The youngster appeared to cope with having to perform a more disciplined role, but Koeman could yet change Barkley’s position again.
“I need to see Ross doing his training sessions and get an opinion on him,” he said, when asked what Barkley’s best position is.
“He will play in one of the midfield positions, but I don’t know where that will be.
“He is a vital young player for Everton.
"At the moment he is playing behind the striker. It’s more of an offensive position than defensive. And I like him as a player.
"He can play either side and has a good view of the pitch.”
Barkley scored 12 goals and made 11 assists for Everton during last campaign.
Everton are likely to have to fend off interest in the skilful midfielder this summer, but they are confident of keeping hold of the prospect.
Barkley is under contract at Everton until 2018, and could be handed a new deal soon.
Barkley is currently at Euro 2016 with England, but he is yet to make an appearance for Roy Hodgson's side at the tournament.