Court officials say a decision could be announced shortly after the conference.
Manning asked U.S. Supreme Court in December to listen to his arguments for a new trial.
Manning has argued that defense attorneys should have done a better job and that black residents were inappropriately excluded from his Oktibbeha County jury. Manning is African-American.
The 5th U.S. Circuit Court of Appeals turned Manning down last July. The court said Manning filed his post-conviction claim too late to be heard in state courts.
In a post-conviction petition, an inmate argues he has found new evidence — or a possible constitutional issue — that could persuade a court to order a new trial.
The 5th Circuit said under state law, Manning had until April 5, 2000, to file a post-conviction petition. Court records show Manning did not file anything with the Mississippi Supreme Court until Oct. 8, 2001.
Attorney General Jim Hood has said he will ask for an execution date to be set if the U.S. Supreme Court denies the request.
Manning, now 44, received two death sentences for the 1992 slayings of two Mississippi State University students, Jon Steckler and Tiffany Miller.
On Dec. 11, 1992, the bodies of Miller and Steckler were discovered in rural Oktibbeha County. Both students had been shot to death, and Miller's car was missing. The vehicle was found the next morning.
Prosecutors said Manning was arrested after he attempted to sell certain items belonging to the victims.