Boss Arsene Wenger has stated he regards regular Champions League qualification as a successful season.
But England midfielder Wilshere wants to look back on a career marked by medals.
It has been seven years since the Gunners lifted a trophy, a stat that bothers the 20-year-old star. He said: "We know how desperate the fans are as well for a trophy and, trust me, the players want silverware.
"When you look back on your career, you don't think: 'Ah right, we finished fourth every year'. You say: 'What have we won?' And that's what you want to do.
"Our aim every season is to challenge for the Premier League which we continue to do so week-in and week-out. We wouldn't be Arsenal FC if we wanted any less."