The Senate could vote on the bill later this week.
On Monday, some families of murder victims got in on the debate.
"A vote for the Racial Justice Act is a vote for us," said Andre Smith, father of a murder victim.
Smith's 21-year-old son was stabbed after an argument at a Raleigh nightclub in 2007. He said he did not support the death penalty.
His son's killer got more than 20 years behind bars and Smith said he was fine with that. He said he believed there was racial bias in death penalty cases and wanted it to end.
"We do believe the person who did the crime, should do the time," Smith said. "We're not saying they shouldn't do the time. What we're saying is that it needs to be fair."
Tom Fewel's said he felt the same way. His daughter was kidnapped and killed while she was on her way to school.
"I'm not here to tell you that any district attorney or law enforcement officer in this state is a racist," Fewel said. "We have evidence that there is racism in this system and this is just one chance the state can take to say are we about to make that mistake again."
Officials said the Racial Justice Act could potentially allow all death row inmates to appeal their case based on racial discrimination.
"What this bill does is allow them to show that someone else has been discriminated in a different county in the district, different district in the state and somehow that proves that they have been discriminated against," House Minority Leader Rep. "Skip" Stam said.
Stam is against the bill and so is the Association of District Attorneys.
They said it would allow white and black death row inmates to claim they were sentenced to death at disproportionate rates compared to other white murderers and black murderers. And it would allow the male inmates to say their sentences are disproportionate to female murderers.
Opponents also said they felt the bill would basically extend the moratorium on executions, while supporters said they hoped so.