The Arsenal skipper sustained an ankle injury in the 5-1 victory against West Ham and was substituted in the 58th minute at Emirates Stadium.
Wilshere took over the captaincy, but Wenger insists he did not tell Vermaelen to hand the role to the 21-year-old.
A leader is somebody who does everything on the pitch to help his team to win. And he does that
“Honestly, Vermaelen made that choice and I let him make it,” the manager said. “It is always interesting to see when Vermaelen is out who he feels spontaneously should be the captain. I did not want to stop him.
“[Wilshere] is naturally a guy who is not scared of anything on the football pitch and that is usually the sign of a leader. He wants to win and shows you that.
“Of course he will be one of the leaders of this club - in fact he already is. A leader is somebody who does everything on the pitch to help his team to win. And he does that.
“If you are asking me if I will take the captaincy away from Vermaelen [then my answer is] no. But I think Jack will be captain of this club one day.
“First of all, because of the quality of the player. Second, his commitment and desire to win. And his understanding of the game. He understands football very well and senses on the pitch what you have to do.”Copyright 2016 The Arsenal Football Club plc. Permission to use quotations from this article is granted subject to appropriate credit being given to www.arsenal.com as the source