The 23-year-old Barkley, who came through the academy at Everton, has been in fine form this campaign, scoring four goals and providing eight assists in the Premier League so far.
"I know the board is talking to his agent. We need to know the answer at the end of the season," Koeman told reporters on Thursday.
"We will offer him the contract that is good for the player. Everton is the best place for him."
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Everton have an outside chance of creeping into the top four and qualifying for next season's Champions League as they sit seventh in the table on 54 points, seven behind fourth-placed Manchester City, who have a game in hand.
"I spoke to Ross a few weeks ago and he mentioned his ambition, Champions League, and I told him I have the same ambition," Koeman added.
"It's normal. Every player would like to play Champions League. Our next step is Europe... Everybody has seen the difference between this season and last, the club is going in the right direction."
Everton will be looking for an eighth consecutive league victory at Goodison Park when they host 12th-placed Burnley on Saturday.
(Reporting by Shravanth Vijayakumar in Bengaluru Editing by Jeremy Gaunt)
