Cole, who defended Terry in court from charges made by Queens Park Rangers player Anton Ferdinand, was reportedly referred to as a "Choc Ice" by a Twitter user.
The term is believed to be mean black on the outside, white on the inside.
Ferdinand's older brother, Rio, who plays for Manchester United, responded to the tweet on Saturday with: "I hear you fella! Choc ice is classic hahahahahaha!!"
Many Twitter users immediately expressed concern about what Ferdinand was implying about Cole, prompting Ferdinand to tweet later that day: "And if I want to laugh at something someone tweets....I will! Hahahahaha! Now stop getting ya knickers in a twist!".
Cole tried to defuse the situation on Sunday by releasing a statement through his lawyers, saying he "has no intention of making a complaint."
"Ashley Cole has been made aware of the discussion following comments appearing on Twitter and wishes to make it clear that he and Rio Ferdinand are good friends ... and Ashley appreciates that tweeting is so quick it often results in off-hand and stray comments."
Ferdinand took to Twitter again on Sunday to explain himself, saying: "What I said yesterday is not a racist term. It's a type of slang/term used by many for someone who is being fake. So there."
Derbyshire Police said the initial tweet that was responded to by Ferdinand is under investigation because of complaints from members of the public.
England Football Association chairman David Bernstein is due to appear before the British Parliament on Tuesday to talk about football governance. But with racism the dominating topic in English football lately, he could be forced to elaborate on further measures the FA could take to eradicate racism from the game.Suggest a correction