67-year-old Vernon Madison was scheduled to be put to death at 6 central time Thursday for the 1985 killing of Mobile Police Officer Julius Schulte.
The court granted Madison a stay of execution until a decision on a petition for a writ of certiorari is reached.
Attorneys had argued that stroke-induced dementia had left Madison unable to remember killing the officer or understand his looming execution.
The state is still waiting for approval to carry out the execution overnight. When Madison was convicted in 1994, the jury recommended life in prison, but the judge in that case sentenced Madison to death.
But the U.S. Supreme Court in November reversed that decision, saying court precedent had not established "that a prisoner is incompetent to be executed because of a failure to remember his commission of the crime".More news: Billionaire investor Soros to Google and Facebook: 'Your days are numbered'
Currently, 182 inmates are serving sentences on Alabama's death row, three of who have been there longer than Madison.
According to court records, Madison killed Cpl Schulte during a domestic dispute between Madison and his girlfriend. Justices will decide whether to let the execution go forward later Thursday.
The son of a slain Alabama police officer said his father should not be forgotten as courts and media focus on the legal wrangling over whether his father's killer should be executed.
In 2016, the Atlanta-based 11th U.S. Circuit Court of Appeals ruled that Madison was no longer legally eligible to be executed because of his memory loss. He requested two oranges as his last meal.
Prosecutors said Madison shot Schulte in the back of his head as he was sitting in his police auto finishing paperwork after responding to a missing child report. They said in their petition that the state failed to disclose that a court-appointed psychologist who evaluated Madison was addicted to narcotics and had been suspended from his practice for forging prescriptions, making his findings invalid. They argued the court should delay the planned injection to review whether executing someone in such a mental condition violates evolving standards of decency and a ban on cruel and unusual punishment.