The Premier League side terminated the 28-year-old's contract this month after manager David Moyes told him he did not have a place in his plans.
The midfielder is searching for a new club and would like to continue his career in another top league in Europe, though he would also consider joining a team in the United States.
"La Liga, Italy or Germany would be a nice change for me. Not many English players have done it," he told Sky Sports. "I always watch La Liga, it was always something I thought about. I like the league, I think it's more technical than the English game. I think it's somewhere I could really show what I can do.
"I don't think I'm too young for the MLS. I think there's been a big improvement in that league. If the right opportunity came up I'd be open to it."
He continued: "I want to play for a club where I feel like I'm an important part of it. I just want to play games - I want to get back to playing 25 to 30 games a season, when you can really get your confidence back and a smile on your face.
"Just playing football, what I used to do when I was a little bit younger. I still feel like I can do it and I want to prove everyone wrong.
"The most important thing for me is finding a club, getting a run of games and playing well. Then you put yourself in the manager's head."
Wilshere's former Arsenal team-mate Mesut Ozil also finds himself in a difficult position, having been left out of the Gunners' squad for the Premier League and Europa League. The England international has sympathy for the former Real Madrid midfielder, but believes he will have a lot of options should he leave the north London side.
"I was in that position. As a footballer you want to be given the opportunity to compete and fight for your place," Wilshere added.
"He's one of the best players I've played with. All I can do is comment on him as a footballer and I know what he can give, so it does make it difficult to understand.
"I'm not in the club, I don't know what's happened. I'm sure a lot of teams in England and Europe, Mesut will be the first name on the team sheet - but that's why [Mikel] Arteta's the manager, he makes the decisions and that's down to him."