An attorney for John Washington Hightower, 63, said in a request for a stay filed with the high court that Tuesday's execution should be put on hold until another appeal of the case is heard.
The attorney, Jeffrey L. Ertel, wrote in the request that he has petitioned for another hearing to ask the high court to review a federal appeals court's denial of a previous appeal in the case.
Mr. Ertel did not immediately return a call Thursday seeking further comment.
In a response filed with the Supreme Court, the state Attorney General's Office argued that the execution should proceed as scheduled, saying there is no basis for a stay.
There was no immediate decision by the high court on the request for the stay, which was filed Wednesday. A separate clemency hearing before the Georgia parole board is scheduled for Monday in Atlanta.
Mr. Hightower's execution for the July 12, 1987, slayings of his wife, Dorothy Hightower, and his two stepdaughters, Evelyn Reaves and Sandra Reaves, at a home in Milledgeville, in central Georgia, was set after previous appeals failed.
If carried out, the execution would be Georgia's first in nearly two years.
Among the evidence investigators said they had against Mr. Hightower: a confession and a flesh- and blood-covered murder weapon found in the car he was driving when he was arrested. His clothes also were stained with blood.
According to authorities, Mr. Hightower admitted he had been having marital problems.
In the admission, he said he had been drinking and snorting cocaine hours before he entered the home where the victims were, placed a gun under a pillow in the room he shared with his wife and waited for everyone to go to sleep.
At about 3 a.m., police say, Mr. Hightower retrieved the gun and shot each of the three victims in the head.
A 3-year-old girl in the house was found unharmed.
Mr. Hightower was arrested about 90 minutes after the shootings while driving his wife's car.