Wilshere will see his current deal with the Gunners expire at the end of next month, and the 26-year-old is yet to pen an extension despite making 38 appearances for the club during the 2017-18 campaign.
The midfielder has been left out of the England squad for the 2018 World Cup, meanwhile, due to concerns over his fitness.
Mills has branded Wilshere "an exceptional player", and feels that he should seek a change of club this summer.
"He's a fabulous talent, an exceptional player, when he is fit," Mills told Sky Sports News. "He needs to be at a club where he plays week in, week out.
"Sometimes you change clubs and it just works, for whatever reason it fits. That hasn't quite happened for him at Arsenal. Why is he constantly picking up these injuries? I think he needs a move and a fresh challenge. But undoubtedly he's a fantastic talent."
Wilshere's number of appearances for Arsenal during the 2017-18 season proved to be his highest total for seven years.