Campbell, who played 73 times for his country, claims the Football Association and the majority of fans do not want a black England captain.
Campbell, 39, who three times captained England in friendly matches, also labelled the appointment of Michael Owen as skipper ahead of him as "embarrassing".
In a new authorised biography serialised in the Sunday Times, Campbell said: "I believe if I was white, I would have been England captain for more than 10 years - it's as simple as that.
"I think the FA wished I was white. I had the credibility, performance-wise to be captain."
Wright and Campbell played together for England in the late 1990s, and while he agrees his former Three Lions colleague may have had the right level of experience, former Arsenal striker Wright suspects there were more suitable candidates for the captaincy.
"I think he's got the credentials to be captain, but to be captain for 10 years? I'm not sure when you see some of the players that were around then," Wright said.
Speaking on BBC Radio 5 Live's 606 show, Wright added: "I want a captain that goes in and gets you going. Sol has never had the demeanour where he is aggressive on the pitch and put people in their place.
"People say David Beckham wasn't like that, but he only had to raise his eyebrows to put someone in their place. Maybe Sol could have grown into it but it's down to the manager who he wants.
"Sol was more than good enough to be a captain but I personally believe there were better candidates at the time."
In his book, Campbell, who first played for England in his Tottenham days, claimed it was down to the FA that he was never a regular England captain.
"I was consistently in the heart of the defence and I was a club captain early in my career," he said.
"I don't think it will change because they don't want it to and probably the majority of fans don't want it either.
"It's all right to have black captains and mixed race in the Under-18s and Under-21s but not for the full national side. There is a ceiling and although no one has ever said it, I believe it's made of glass."
On the appointment of Owen, who wore the armband on eight occasions, he added: "I think the FA didn't want me to have a voice. Owen was a fantastic forward but nowhere near being a captain.
"It was embarrassing.
"I've asked myself many times why I wasn't. I keep coming up with the same answer. It was the colour of my skin."
It is understood the FA is not planning to respond directly to the allegations.
Former FA executive director David Davies, who was at the organisation when Campbell was a player, said he was "surprised" by the former defender's allegations and believed they were "wrong".
"I am surprised by this because the reality is that in my time the managers selected the captain," Davies told BBC Radio 5 Live's Sportsweek programme.
"I never - and to my knowledge nobody else in the FA - ever sought to influence them.
"My belief is (the allegations) are wrong. I also have to say I am quite proud, along with a number of people of that era, of setting up (anti-discrimination campaign group) Kick It Out in football. Progress was made, I would love to talk to Sol about this story and this concern and this upset he has about why he wasn't the England captain.
"He clearly is upset about that time and he feels that perhaps he should have been, he would believe, more seriously considered as the England captain."