In March, Murdaugh was convicted of shooting and killing his wife, Maggie, and youngest son, Paul. They were shot multiple times. Murdaugh is currently serving a life sentence.
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His lawyers already filed notice they planned to challenge his guilty verdict, but haven't yet laid out their legal argument.
In an interview just this week, Murdaugh attorney Jim Griffin said as part of their appeal, his team interviewed jurors in the case, looking for bias.
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"There are serious questions as to whether this jury was subject to outside influences during the case," Griffin said.
Trial attorney Danielle Cohen Higgins weighed in.
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"Appeals are extraordinarily difficult to win. The question that I'm sure the counsel for Alex Murdaugh and this appeal is going to argue is, 'There's no way that this small town in South Carolina where this family had such a prominent name, there wasn't a person in the area that didn't know the Murdaugh name,'" Higgins said.
Similar claims about an allegedly tainted jury were made by Murdaugh's eldest son, Buster, in a new Fox Nation docuseries, "The Fall of the House of Murdaugh."
Breaking his silence, Buster also said he doesn't think his dad killed his mother and brother, but added that he believes his father has some psychopathic qualities.
"I'm not prepared to sit here and say that it encompasses him as a whole, but, certainly, I think there are characteristics where you look at the manipulation and the lies and the carrying out of that such, and I think that's a fair assessment," Buster said.