Wenger extended his stay at Arsenal until 2019 when he signed a new contract at the club last Wednesday.
The Gunners go into next season without Champions League football for the first time in 20 years after finishing a disappointing fifth in the Premier League.
They did beat Chelsea 2-1 in the FA Cup final seven days ago but Wilkins believes that performance should not disguise the problems they have.
"They've got Europa League, so they will be playing Thursday, Sunday which I always think is a bit of a bug-bite, because you're almost playing catch-up all the time," the former Manchester United midfielder said on Sky Sports.
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Sead Kolasinac – 4.5
Ray Wilkins has backed Arsenal to have a big transfer window
He's done a wonderful job for Arsenal and for English football, I just hope he puts his, not his hand in his pocket, but the club's money up front
Ray Wilkins on Arsenal boss Arsene Wenger
"I don't think their fantastic performance at the weekend, and it was fantastic and one of the best cup finals I've seen in years, should cover up the cracks that we saw from December through to the end of February, March.
"Because they weren't good enough and we can't allow that to cover up what was a pretty poor season for them, because they should be top four."
Arsenal have reportedly had a bid of £87million turned down for Monaco teenage sensation Kylian Mbappe and Wilkins believes that would be a great piece of business.
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"I think Arsene will go out and I hope he spends money because I don't like seeing someone of his stature being ridiculed as he was last year," he added.
Ray Wilkins believes Kylian Mbappe would be great business for Arsenal
"He's done a wonderful job for Arsenal and for English football, I just hope he puts his, not his hand in his pocket, but the club's money up front and buy some real stellar people.
"If that's the going rate for an 18-year-old, you've actually got a sell-on.
"He could play for you for five years and still only be 23, then he could go to Barcelona for £150million, so that's not bad business.
"If he wins you the Premier League, that's good business and that's the way it has to be."