The former Arsenal and Tottenham Hotspur defender, amassed 73 caps between 1996 and 2007 and led England out on three occasions.
His allegations were revealed in extracts from his biography which is being serialised in a newspaper.
Graham Taylor, who was in charge of England between 1990 and 1993, said: "Sol is making a real go of giving his book a good sell. I cannot go along with what he is saying.
"I never had any influence on me to select who should be captain regardless of the colour of his skin."
Campbell, 39, says in his book: "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 alright to have black captains and mixed race in the Under-18s and Under-21s but not for the full national side. I think the FA wished I was white.
"I had the credibility, performance-wise, to be captain . . . and I was a club captain early on in my career [at Tottenham]. I believe if I was white I would've been England captain for more than 10 years."
Taylor highlighted midfielder Paul Ince as England's first black captain.
"No-one from the Football Association ever gave me an impression that I shouldn't have given the captaincy to him," the former Aston Villa and Watford manager said. There was none of that in my managerial career."
Sven-Goran Eriksson, who gave Campbell the armband against the United States in 2005, said there was "not a chance" the claim were true.
Campbell alleges it was "embarrassing" that Eriksson made Michael Owen captain ahead of him. He adds: "I've asked myself many times why I wasn't [made captain]. I keep coming up with the same answer. It was the colour of my skin."
However John Barnes, a trailblazer for black England players, believes the other captains during the time Campbell was playing for the national side deserved the honour.
"I don't think Sol would have been captain for 10 years because if you look at the captains - they were Tony Adams and then after that Alan Shearer," Barnes said.
"Then David Beckham became captain and that was a PR exercise for the FA. I don't think necessarily it was because of the colour of Sol's skin."