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Henry Curtis Jackson was convicted in 1991 of killing his two nieces and two nephews.
Jackson's attorney, Robert Davis Jr. of Tupelo, filed a clemency request Friday. Davis said 47-year-old Jackson would remain in prison if he's not executed.
"Life without parole is our best hope," Davis said.
Davis said he has filed appeals on Jackson's behalf up to the U.S. Supreme Court, and is unlikely to file a new appeal at this time.
The killings happened on Nov. 1, 1990, in Leflore County in Jackson's mother's home. His mother was attending church at the time. Jackson was also found guilty of stabbing two other relatives, both of whom survived.
Jackson, who was 26 at the time of the deaths, turned himself in to the West Point Police Department on Nov. 5, 1990. He was convicted in September 1991 of four counts of capital murder for killing the children, who ranged in age from 2 to 5 years old.
Republican Bryant has not granted clemency requests to previous inmates, though his spokesman, Mick Bullock, said Bryant is reviewing Jackson's case.