The 26-year-old has endured several injury-plagued years and was widely expected to leave the club, most notably after joining Bournemouth on loan.
However, Wilshere's Gunners career appears to be back on track with 20 appearances in all competitions this term, 11 of these coming in the Premier League.
Wilshere, whose contract runs out in the summer, also captained Arsenal against Chelsea in their EFL Cup semi-final first leg at Stamford Bridge, which finished as a goalless draw.
"I want to be Arsenal captain," the Englishman told Sky Sports News. "I really enjoyed the other night, it was my first start as captain and walking the boys out was a special feeling for me and my family so that is something that I want to do. I never really stopped dreaming about that.
"There was a time last year when Arsenal were playing against Bournemouth at the Vitality where I sat in the crowd watching and thinking 'am I going to go back to Arsenal and play?' but apart from that, I never really thought about not becoming the player everyone thought I would be or not fulfilling my potential.
"I just wanted to prove my fitness at Bournemouth then come back here and try and get into the team. I played the long game and I worked hard. I'm not saying that's over but I'm the team now and I need to work even harder maybe to keep my place and improve as a player as well.
"There's not a decision to make in January. I've still got six months left of my contract, I'm an Arsenal player and I'm enjoying playing week in, week out. I've had loads of questions about my future and there's nothing more I can say until something's official and we can concentrate on the future but right now, I'm concentrating on each game as it comes.
"I've just got back into the team and the club, I suppose, were just waiting for that moment where I've proved my fitness and I've proved that I can play in the Premier League each week for a top club so now that's happened.
"We're not waiting for anything really, maybe the right time when there's a little break in the fixtures - which there's not - but I'm sure in a couple of weeks, something will get done or the boss will say something and we'll go from there."
Wilshere will be back at the Vitality Stadium on Sunday when Arsenal face the Cherries in the Premier League.