The Gunners splashed out a club-record £42.5m to bring the Germany international to north London last September but former Arsenal captain Adams says that Wilshere should have been offended by the move.
“He must show more hunger and professionalism. He’s in danger of not fulfilling his potential and his career could be over in the blink of an eye,” said Adams in an interview with The Sun.
“If push came to shove, for a cup final or big match, Arsene would pick Ozil ahead of Jack.
“He should have taken it as a personal insult that Arsenal went out and signed Ozil when they already had the same talent at their disposal in him.
“He has to be more verbal, shake things up and show more fire in his belly.”
Wilshere, 22, scored three goals and made four assists in 24 Premier League games for Arsenal last season.
MORE: Man United latest news
MORE: Arsenal latest news
MORE: Chelsea latest news
MORE: Liverpool latest news