Gregory Mills, a veteran of Florida's death row, is scheduled to die by lethal injection May 2 for the fatal shooting of James Wright in Wright's Sanford home.
The 70-year-old man was shot as he came out of his bedroom during an early morning break-in on May 25, 1979.
Only 10 of the 371 people on Florida's death row have been there longer than Mills, who is now 43.
Todd Scher, Mills' attorney, told Florida's high court in oral arguments Tuesday that the man who testified against Mills two decades ago in exchange for immunity may have been the killer.
That man, Vincent Ashley, has told a prosecutor that he doesn't know what he might say if brought again to the stand - and might say he was the one who shot Wright.
Kenneth Nunnelley, the assistant attorney general defending Mills' death sentence, told the state Supreme Court that Ashley was only expressing his unwillingness to testify again.
Also Tuesday, Scher filed an appeal with the U.S. Supreme Court, asking the nation's high court to review a 4-3 decision issued by the Florida Supreme Court earlier this month.
In that decision, the majority refused to stop Mills' execution but the dissenters, in a sharply worded opinion, said the execution of Mills would not only be an injustice but also an arbitrary application of the death penalty.
The issue in that appeal is that the jury that convicted Mills recommended that he be sentenced to life in prison. The trial judge sentenced Mills to death and in 1985, the Florida Supreme Court upheld that sentence.
Under Florida law, judges are supposed to abide by jury recommendations for life in prison unless "no reasonable person" would agree with the jury recommendation.
Since 1989, Florida's high court has rarely upheld a death sentence in a case where the jury recommended life.