Davis was convicted of the 1989 murder of Savannah police officer Mark MacPhail, but his execution has been delayed three times over concerns that he is not the killer.
Supporters say he deserves a new hearing because seven of the trial's witnesses have recanted or backed off parts of their testimony.
The 11th U.S. Circuit Court of Appeals cleared the way for the execution in April, ruling it was "unpersuaded" by the affidavits. The U.S. Supreme Court could decide as early as Thursday whether to hear the case.