Campbell branded the FA "institutionally racist" in his eponymous biography, the first part of which appeared in serialised format in yesterday’s Sunday Times.
The former Arsenal and Tottenham Hotspur defender, who played 73 times for his country and was captain three times, has claimed that he would have been first choice had he not been black.
"I believe if I was white I would've been England captain for more than 10 years," he wrote. "It's as simple as that. I think the FA wished I was white."
The FA declined to comment, but one source, who firmly denied the allegations, said it was considering whether to do so, adding "a lot of people are speaking on the subject on our behalf".
Dave Wilson, group MD of Bell Pottinger, warned the FA against reacting to the allegations.
"These are one man’s claims and do not deserve a response," he said, stating that the "misguided" claims would not gain wide traction.
"The FA cannot respond to every individual claim or point and it is notable that other players have not yet come out in support of him."
Last year the FA was criticised by its only female board member Heather Rabbatts for selecting an all-white, all-male commission to help improve the England team.
However, Wilson said: "Though there is clearly some progress that needs to be made the FA has made great strides in helping eradicate racism in football.
"This claim [about the captaincy] is just not one that many people would endorse."