The message was posted on January 31 – a day after Dyer parted company with the Scottish Premiership side.
A Police Scotland spokesman said: “A 43-year-old man has been charged in connection with offensive communications posted online. A report has been submitted to the procurator fiscal.”
Kilmarnock say fans reported the message to the police and it was the second time in a month that 55-year-old Dyer had been the subject of racist abuse.
Dyer received a letter containing racist abuse at the club’s Rugby Park home at the end of December.
He said at the time: “I don’t mind someone telling me that they want me to leave the football club because I’m not doing a good job. I can take that.
“But not to tell me about the colour of my skin, that someone else is better than me or better than my race. It’s not right.
“It hurts but it’s not Kilmarnock supporters, not Kilmarnock Football Club.
“They are not about that. They have been very good to me since I have been here.”