Southgate, who was named permanent England manager last November, has left Rooney out of the forthcoming games against Germany and Lithuania even though he could return from a knee injury for Manchester United this weekend.
Rooney has worn the England armband for the last two years, but has struggled for form and fitness this season.
The 31-year-old has become a fringe member of Manchester United's first team, while Southgate chose not to start him in a World Cup qualifier against Slovenia in October.
Rooney's last outing with England was in the 3-0 win over Scotland four months ago. But it was overshadowed after he was accused of drinking heavily with hotel guests at the team hotel following the match.
Southgate is considering changing the nature of the role and explained: "We have this thing about 'an England captain', but really the captain is the person that is captain in the next game, isn't it? Or the game on the next day.
"The key for me is how do we develop more leaders? It's not something I think too much about really.
"I guess there's almost a figurehead position and then I have to make a decision for every game based on who's playing. You might have that at a club, but international's different."
When asked if that meant Rooney doesn't just keep the title of 'England captain', he added: "I don't quite know who gives it and what it means. I think it's a really good question and probably something I need to go and have a think about!
— England (@England) March 16, 2017
"I always just assume you pick a team for a game and the captain of that game is the captain. I think that's why you need a leadership group because if a player is injured and your captain's not around, who comes and knocks on the door and asks for jelly babies in the afternoon or whatever it is?!
"For me that's why you need a group of not just senior players, because some younger ones could go in and start to drive some of the conversations and the culture around the team.
"But it's probably something moving forward that's one of the areas we have to think about."
Rooney's lack of starts for Manchester United - he has been named in the first XI for the Premier League just nine times - has put his place as an England starter under threat.
Significantly, Southgate named Dele Alli, Adam Lallana and Ross Barkley as players who can be picked in the No.10 role that he plays.
"Every player needs to feel that competition for places. That's where we are as a team. I don't think there's many positions, if any, where you could say, 'Player X is guaranteed his position.'
"We'll only have a successful team if we've got that competition for places because that drives performance. For the time being, I'm happy for that to be the case.
"We have to look at Wayne as a number 10, which is his predominant role. In the last two games we've played Dele there, we've played Lallana there.
"Both are playing very well, scoring and assisting for their clubs. Ross Barkley has been playing very well for his club. So there's competition. I can't dress that up any other way.
"There are some very good players and it's a battle to get in this squad. Wayne totally understands that. He's the most realistic senior player I think I've dealt with in terms of how he views the game.
"He's got an intelligent view of it. He doesn't have any expectations of being treated differently or treated in a special way. Whenever I have those conversations with him, I'm really impressed."
Rooney's future at Manchester United has also come under a lot of scrutiny and he has already rejected a move to the Chinese Super League in January.
He is expected to be approached again in the summer over a move to the Far East, while MLS side LA Galaxy is an option. Former club Everton are also keen.
A move to China and America is likely to curtail his England career a year before the World Cup in Russia.
Southgate refused to confirm that will be the case but refused to deny it either: “Until anything like that happens it’s not something I need to think about really.
"I guess for all of them the first point is to be playing as regularly as possible. That’s the key as far as I can see.
"I don’t know enough about the (Chinese) league. I’ve not seen the games so it would be unfair of me to judge."
Meanwhile, Rooney has been invited to attend a conference session with the rest of England's players and staff on Monday where Southgate will explain what he has planned for the next 18 months.