That is the according to former Three Lions skipper Bryan Robson.
Southgate has opted for a rotation of the armband recently, giving the role to Cahill, Harry Kane, Jordan Henderson and Joe Hart in fixtures since Wayne Rooney's international retirement.
England are preparing for their final two World Cup Qualifying clashes against Slovenia and Lithuania this week, with no decision on who will lead the national team out.
There has been a lot of clamour for Spurs striker Kane to be given the honour, after his scintillating form in the month of September.
But Robson isn't in favour of the forward being named skipper.
"I have never been an advocate of strikers being captain," Robson wrote in the Daily Mail.
"There is a clamour for Harry Kane to lead England and I understand that. He's scoring goals, full of confidence and works his socks off but I wouldn't make him captain.
"It needs to be a centre half or midfielder as they are closer to the main action and in a better position to assess the team shape.
"The important basic in football is to stay switched on, it's all about what you do next.
"When a striker misses a chance there's a moment when his head may go down or he ponders what he should have done.
"Behind him, the team need to be in position to regain possession and a centre half is best placed to shift people into place."
Robson believes experienced Chelsea man Cahill is therefore the right man to be named captain.
"That's why, for me, the captain should be Gary Cahill," he added.
"I had Gary for a three-month spell on loan at Sheffield United when he was 21. Even then he was a good talker and organiser, and he's a quality defender.
"He's worked alongside one of the best captains in John Terry and has taken over his mantle at Chelsea.
"He's got experience, 57 caps, and when he talks to younger players he will command their respect."
England need a win to ensure qualification with one game to spare, or a draw if other results go in their favour.
