Kenneth Fults was sentenced for the 1996 killing of Cathy Bounds who was shot five times in the back of the head.
Signed statements by white juror Thomas Buffington show that he used racial slurs when referring to Fults.
So far state and federal judges have rejected the appeal.
Fults pleaded guilty at the 1997 trial and the jury sentenced him to death.
At the trial Mr Buffington told judges and lawyers on both sides that he harboured no racial prejudice.
Eight years later, an investigator for Fults spoke to Mr Buffington about his experience on the jury. Mr Buffington who was 79 at the time of the interview, used racial slurs to describe Fults.
According to the signed, April 12, 2005 affidavit, Mr Buffington said, "Once he pled guilty, I knew I would vote for the death penalty because that's what that [N-word] deserved."
State judges argue Fults waited too long to present the statement from Buffington and did not explain why the evidence couldn't be found sooner.
However, lawyer Lindsay N. Bennett, who is representing Fults, said it is common in Georgia for a defendant's legal team to reach out to jurors at that stage of an appeal, but not earlier.
"At this point, he has essentially reached the end stages of his legal proceedings with no court having assessed the substantive evidence in this case," Bennett said.
A black inmate has appealed to the US Supreme Court for a new sentencing hearing, arguing one of the jurors who sent to him to death row was racist.