Adams won four league titles and three FA Cups during his time at Arsenal, with the honours secured by the London club gradually drying up following his retirement in 2002.
However, Wenger can bring an end to Arsenal's nine-year trophy drought on Saturday in the FA Cup final against Hull City.
|16/1||Arsenal are 16/1 with Bet365 to win the FA Cup final on penalties|
With Wenger likely to sign a new contract following the final, Adams feels it is European glory that drives the Frenchman to remain at the club amid criticism of a perceived lack of progress in recent seasons.
"He really wants to win the Champions League. And he still thinks this group of players can do it. While he's got that belief, he'll stick around," Adams told reporters.
"If he thought that this squad couldn't win him the Champions League, he would move on to Barcelona or a club that would deliver that for him.
"Maybe he comes across arrogant at times. I don't share his optimism about his squad. But he should stay.
"Financially, he does so well for the club. There's a great structure in place, and maybe it is better the devil you know."
Adams also drew likeness between this year's trophy-chasing Arsenal side and that of the successful League Cup side in 1987.
He helped see off Liverpool that year under George Graham as they picked up silverware for the first time since 1979 and the former England defender feels Wenger's charges must take the opportunity against Hull.
"It's about getting the job done. None of this Arsenal team knows what it's like to win a trophy for the club. But it's their time and they have to grab these moments," he continued.
"If you've done it before, there are moments where you just sense the opportunity. I'm not so sure this group of players have that winning mentality yet."