Court officials said a decision could be announced shortly after the conference.
In one case, John Christopher Blakeney is seeking a new trial after his murder conviction in Jones County.
The Mississippi Supreme Court declined to hear his appeal earlier this year. The state Court of Appeals upheld his conviction in 2009.
Blakeney was convicted for his role in the murders of his wife's grandparents. He was sentenced in 2007 to life in prison for the slaying of Willie Kitchens on July 10, 2006. A Jones County jury was unable to reach a verdict in the murder of Anita Kitchens.
Blakeney's wife, Wanda, who was tried separately, was convicted of two counts of murder and sentenced to two consecutive life terms. She was not a party to her husband's appeal.
Blakeney claimed he was in a hotel room in Denham Springs, La., at the time of the killings and had received a call on his cellular telephone from his wife around the time prosecutors said he was at the scene of the killings.
Prosecutors said Blakeney's phone had a feature that allowed the phone to answer by itself. They said Blakeney had left the phone in Denham Springs and called the phone from Jones County to set up an alibi.
In the second case, Lonzie Earl Nichols is seeking a new trial after his 2006 conviction in Clay County for murder involving a deadly car crash in 2004.
Nichols was sentenced to life in prison. The sentence and conviction were upheld last year by the state Court of Appeals.
Prosecutors said Kiki Johnson and her unborn child were killed on July 24, 2004, when the car in which they were riding collided with a vehicle driven by Nichols.
The wreck occurred about four miles south of West Point on U.S. Highway 45 Alternate.
Nichols claimed the evidence did not support a murder conviction. The Mississippi court said there was evidence that Nichols' deliberately rear-ended the Johnson car, an action which contributed to the wreck.