The England international retweeted two posts following his team's 3-0 loss to Swansea City on Saturday afternoon.
Both posts criticised Cole's performance, as well as using a racist term to describe the 28-year-old.
Cole posted on his account: "Listen, i take racism a lot lighter than others and i do understand the banter that comes a long with it to get under peoples skin but...
"...but just say that, whether i am crap or had a bad game has nothing to do with my race, creed or religion. lets just keep it FOOTBALL. Kapeesh?"
Cole has been racially abused previously, during a Carling Cup match against Millwall in 2009.