The under-fire Frenchman has been at the Gunners for nearly 20 years.
The Frenchman has come under fire this season as the Gunners look set to miss out on what is widely considered the most winnable Premier League title perhaps since the competition's inception in the early 90s.
Wenger, whose last title came 12 years ago, has been the subject of protest from a section of the club's fans in recent weeks and his future at the club remains unclear.
The 66-year-old will celebrate 20 years in the job in September if he remains in charge for next season, and Bergkamp, who won seven trophies alongside Wenger at Arsenal, is adamant that his former boss should not be sacked.
"I still think he’s got that drive and if he gets the opportunity to stay there, he will take it I’m sure. He’s still a good manager and you never know what you’ll get if you bring a new one in. That’s the problem," he said to The Mail.
"Look what happened at Man United when Sir Alex Ferguson left. You never know what’s better.
"To bring someone new, someone fresh or stick to someone who knows the club. I’m sure (Wenger) is still ambitious to win trophies. As long as you don’t lose the connection with the team."
Wenger has not helped his cause in the other competitions, with Arsenal, the winner of the last two FA Cups, surrendering that title with a defeat by Watford, while Barcelona eliminated them from the Champions League at the last-16 stage - again.
Thirteen points behind leaders Leicester City, third-placed Arsenal host Crystal Palace on Sunday evening.