The 31-year-old winger is believed to be having personal problems across the Channel and could be interested in making a return to the Aviva Premiership in a bid to rekindle his England career ahead of the World Cup next year.
Boudjellal has admitted there is currently some doubt regarding Ashton's future but hopes the club can convince the former Saracens and Northampton Saints flyer to remain with Toulon.
"[Ashton] asked to leave the club, then he asked to return," Boudjellal said during a Facebook Live chat.
"We hope that Chris Ashton will stay in Toulon. We are currently in discussion with him. We are in the midst of personal problems. There is no ultimatum about it.
"There are different issues - there is the family problem and there is the issue of selection. We want him to stay. The ball is in his court now."