Barkley's current £65,000-a-week deal expires at the end of next season and the midfielder is yet to reach an agreement with Everton despite the offer of fresh terms and being handed a deadline to commit which has now passed.
Tottenham have been monitoring Barkley's situation and Everton have bolstered in the England international's position with the £24million arrival of Ajax captain Davy Klaassen on Thursday.
"I think Ronald put his cards on the table with Ross before the end of the season and he gave him an ultimatum," Stubbs told talkSPORT.
"The fact that we are three weeks gone and Ross hasn't signed, then I think that's the biggest indicator that Ross' future is going to lie elsewhere.
"The one thing that [Koeman] said towards the end is that he wanted more creativity in the team and I think, with Klaassen, he's certainly got that - his record is very good and he's a great age."
