The Gunners face a Champions League trip to Bayern Munich on Wednesday, before an FA Cup clash with Sutton next Monday.
Arsene Wenger's side then travel to Liverpool for a crucial Premier League match before the return fixture with Bayern.
And Mills believes Arsenal could struggle in these matches due to their lack of character.
Danny Mills believes Arsene Wenger's Arsenal could be out of all three competitions in a month
When asked about Arsenal's upcoming fixtures, Mills told Sky Sports: "It's hugely important.
"Of course Arsenal finished top of their group but drew Bayern Munich – that's just bad luck.
"Sutton won't be easy, I was there when they played Leeds at home on the astroturf. It makes a big, big difference and it won't be easy for Arsenal who you expect to make changes.
"If they were to lose against Munich, lose to Sutton, lose to Liverpool then suddenly they're out of all three competitions.
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Mon, February 13, 2017
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"In my eyes I don't think they are really in the Premier League race at the moment.
"But it could be a really disastrous next four or five games for Arsenal."
"Do they have the character? Do they have the mental resolve to get through this really tough next month of fixtures?
"I'm not sure they do and I'm not convinced Arsene Wenger thinks they do either."