The England international is reportedly set to become the club captain, but Savage thinks that the lack of silverware at the Gunners in recent years may cause the midfielder to move on.
"I think they risk losing Jack Wilshere to a bigger club capable of fighting for major prizes, just as they have lost Robin van Persie, Cesc Fabregas, Thierry Henry Patrick Vieira and others down the years," Savage told The Mirror.
"Terminal decline is probably a bit harsh, but I fear Arsenal's season could get even worse.
"Now they have barely a dozen games left to try and salvage fourth place in the Premier League – but don't put your mortgage on it. They are going to struggle to make the Champions League again next season."