The Gunners have endured much criticism for their poor start to the Premier League season, and a lack of leadership on and off the pitch has troubled manager Arsene Wenger.
According to reports in talkSPORT, the 21-year-old Pole has stepped up and put his name up for the job, looking to take over from current club captain Robin van Persie.
This is despite him being one of the younger players in the current Arsenal set up, as Szczesny believes he has the authority to command his team-mates on the pitch.
Speaking to polskathetimes.pl, the confident custodian spoke out, saying: "I feel that when I say something, everyone listens to me. More so on the field than in the dressing room, but that's more important I think.
"I want to be the captain of the team. I feel that I am able to take on that responsibility. I always wanted to be captain and if I ever get that honour, I will gladly put on the armband."
The Pole was also frustrated at the Gunners' 2-1 loss to Tottenham Hotspur last weekend, despite his best efforts.
"If you make good saves you want to take something out of the game," he said. "I made a few good saves but it gave nothing to the team because we lost the game and came out with no points so it doesn't matter really", he said to his club's website.
"I think in the second half you could see an improvement but we were a little bit too sloppy on the ball and they looked dangerous on the counter attack. In the second half we scored, controlled the game but then gave a silly goal away."
He did admit that Tottenham's first goal, scored by Rafael van der Vaart, was worthy of praise, despite accusations of handball in the buildup to the Dutchman's finish.
"I didn't see the handball, I thought he controlled it and finished very well," said the Pole. "It was a good finish.
"The second goal I was very disappointed in myself - I shouldn't get beaten from that kind of distance so I will take that myself.
"There was a lot of movement on the ball and I reacted too late. I got a touch on it but it wasn't strong enough. I blame myself for the second one but other than that I didn't think the team deserved to lose the game", he added.
The Pole also admitted that he could very well leave Arsenal for Barcelona one day to follow Cesc Fabregas' footsteps.
"I can't declare that I will spend my entire career here, but if I leave I will have a very important reason or I will go to Barcelona.
"I can't imagine a situation where I would leave this club just for money", he concluded.