Tottenham Hotspur are one of the clubs being linked with Barkley, who has one year left on his contract.
The England midfielder has just over a year left on his current deal and several months ago Koeman set a deadline of the end of the season for the 23-year-old to re-sign or be sold.
If the England worldwide plumps for Chelsea, he will likely see considerably less game-time but would profit from the incredible offensive talents that will surround him when he does see the field of play. Next week we need an answer, if we go for a holiday, he will [either] accept a contract or we will sell the player.
Should Barkley, who is one of English football's most high profile names, leave Goodison Park, Everton will undoubtedly be looking for a replacement.
"If you take so much time [to sign], you have doubts [about staying]. At the moment I want my players concentrated on the next three games".
Koeman disagrees with the Belgian striker but believes he can still improve his game. He knows what he needs to improve.More news: Highlights of the latest hearing on Russian election meddling
Asked whether a conclusion to the deal was close, Koeman said: "Yes". We still have three very hard games left.
"If you see the performances of Rom overall in the season, it's better than it was".
"We had talks about the future, about ambitions, but that's between us".
Though Everton will be keen to keep hold of Barkley, the chance to play in the Champions League and challenge for Premier League glory next season with Spurs may be appealing.
Everton face Watford on Friday before finishing their Premier League campaign at Arsenal on Sunday, 21 May.
"We've had some great wins - we've had a great home season", he said.