England's number one has been in sensational form for City this season, and has been rumoured to be in contention for the captaincy himself.
But the 24-year-old believes his Premier League rival Rooney deserves the armband against San Marino.
Speaking to Sky Sports, Hart said: "I've not thought about it at all. How the captain is chosen is the manager's decision. I've got a big responsibility to keep my place in the team for club and country.
"If I do get the nod, I'll try and give as much as I can in terms of helping the team. But I think it's between Wayne Rooney and Wayne Rooney if we are going to go on experience and how captain have been chosen in the past."
Regular England captain Steven Gerrard is suspended for the Group H game against San Marino on Friday, and there are doubts over the fitness of next-in-line Frank Lampard.
Manchester United forward Rooney has captained England once before, and the Old Trafford striker has 76 caps for his country, despite being still just 26 years old.
The United forward is expected to link up with Manchester United teammate Danny Welbeck for England, as Roy Hodgson's side look to get back to winning ways after their 0-0 against Ukraine last month.
Manchester City keeper Joe Hart has been one of the stars of Roberto Mancini's team this season, and was particularly impressive against Real Madrid in the Champions League. But the England goalkeeper has been called into action more than Manchester City would like, as their normally-watertight defence has wobbled in recent weeks.
The City shot-stopper has just 24 caps for England, but his impressive displays have led to calls to hand him the skippers' armband already.