The 23-year-old's future has been up in the air since the tail end of last season when his manager insisted that he make a decision over whether to sign a new contract be made by the end of the campaign.
That deadline passed without any movement by Barkley, leading to increasing speculation that he could leave Everton with 12 months remaining on his current terms.
Tottenham Hotspur have been credited with serious interest in the midfielder and Arsenal and Manchester United have entered the conversation in recent days as talk in the media continues but there have not been, as yet, any firm offers for Barkley.
That could force the England international to sit out the final year of his contract at Goodison and leave as a free agent next summer, although Koeman will be hoping that he could move him off the books before the transfer deadline for a fee after he has recovered from groin surgery.
"We made a good offer to him to sign a new contract," Koeman said according to the Daily Star. "He declined and told me he is looking for a new challenge."
"What I heard from the board is that there's not really an offer on the table for Ross.
"I'm looking to other players and that's my priority —- not Ross.
"It's his decision and everybody knows what the situation is.
"We work with respectful people and he is back after his surgery then he will be part of first-team sessions."