But the former Celtic full-back believes the Parkhead power-brokers have a "duty" to take the decision out of the manager's hands after their remote title hopes crumbled further.
A 2-2 draw with Livingston on Wednesday night left Celtic 20 points adrift of Scottish Premiership leaders Rangers with just two games in hand now.
Lennon insisted afterwards he "absolutely" would not walk away, adding: "I've put too much into this. Too much of my life. And a month ago we won the treble.
"You keep asking if I'm going to be here. I will be until I'm told otherwise."
The outcome of a review of the manager's progress is expected before Celtic's next game, at home to Hamilton next Wednesday.
Wilson told Sky Sports News: "To be honest I didn't expect anything less from Neil. I think a lot of people are calling for him to resign but, knowing Neil the way I do - I played under him, I played with him, he's a fantastic leader and he's a fighter, he's a scrapper - he won't quit.
"And I certainly didn't expect anything less from his comments.
"But I do think there's a duty of the club from the people above him to make the decision.
"When you look at the results recently, 22 games and seven wins, or eight if you include the Scottish Cup final on penalties, for any Celtic manager that's a tough run of results to survive.
"I think Neil will know even himself that historically that hasn't been good enough to keep other Celtic managers.
"I was here under Tony Mowbray, I played a part in that. I know fine well that the axe falls pretty quickly.
"And when you add the run of results to the other calamitous things that happened this season, you can understand why the Celtic fans are calling for change now."
The Celtic board twice publicly backed Lennon late last year with the second vote of confidence stating his job was safe pending a review in the new year.