"I'm talking to the club and I probably will be committing my future to them in the next couple of weeks," Sunday newspapers quoted the 20-year-old as saying.
"I know the team are going through a bad spell but that's like any team," he added. "We can get back to being the force we were."
Wilshere said people were quick to criticise Wenger for the north London club's slump this season but the players, who play at bottom side Reading on Monday, had to stand up and be counted.
A nearly full-strength Arsenal were knocked out of the League Cup by fourth-tier Bradford City on penalties on Tuesday.
The club have not won a trophy since the 2005 FA Cup and are eighth in the League, albeit only three points behind fourth-placed Everton and with a game in hand, with Wenger facing increasing criticism.
The Frenchman has seen a steady exodus of top talent, including Cesc Fabregas, Robin van Persie and Samir Nasri, with England winger Theo Walcott recently rejecting a new five-year deal with the club.
Wilshere, who missed the whole of last season through injury and returned only in October, said Arsenal could still win the FA Cup and Champions League and strongly defended Wenger's record.
"Every year he has been in the Champions League and there is no other manager who has done that," he told Sky Sports News. "He has built a legacy at this club. He changed everything from when he came in 15-16 years ago.
"He has also been great for me. He has shown great faith in me from when I was young. He stuck with me when I was injured and has still stuck by me. For me, there is no question he's the right man."
Wilshere said that one day he would like to captain the club he joined as a nine-year-old, even if he was not yet ready for the role. (Reporting by Alan Baldwin, editing by Clare Fallon)