The Frenchman, who will celebrate 20 years in charge of the Gunners this September, will come to the end of his current contract when the 2016-17 season concludes.
"[Wenger's] going to see out his contract," Smith told Sports Mole. "I would be surprised if the club offered him another one. I'd be surprised because if the mood is as it is at the moment in a year's time, the fans wouldn't receive that favourably, so you have to think of that.
"He's been there a long time, so hopefully next year under Arsene they can go one step further. [Winning the league] would be a great way to bow out if that's what he chose."
The North London outfit have not tasted Premier League glory since 2004, but they have won the FA Cup twice in the last three seasons.
Alan Smith was speaking at online bookmaker Betway's Euro 2016 Kick-Off Show. Watch the video and read more in-depth Euro 2016 opinion at blog.betway.com.