Richard Glossip was set to die by lethal injection at 3:00 pm (2000 GMT) following a two-week reprieve to allow time for the consideration of new evidence in his murder conviction.
His execution would be the second in as many days in the United States, after Georgia yesterday went ahead with an execution -- the first of a woman in 70 years in the southern state -- despite a similar call for clemency from the pontiff.
He was subsequently convicted based on the testimony of Sneed, who pleaded guilty and was able to negotiate a life sentence by claiming his co-worker had masterminded the plot.
In a September 21 letter to state Governor Mary Fallin, the pope's representative asked that Glossip's sentence be commuted, writing it would "give clearer witness to the value and dignity of every person's life," broadcaster NBC reported.
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In a speech to the US Congress during his six-day tour of the United States last week, the pontiff had called for the global abolition of the death penalty.
Glossip's case has also attracted other high-profile attention, with actress Susan Sarandon and billionaire Richard Branson among those who have voiced concern.
Branson had appealed to the public to call Fallin's office and request a fresh 60-day stay of execution. "To the Government & citizens of Oklahoma: your State is about to kill a man who may well be innocent," he tweeted yesterday.
Glossip's lawyers produced new evidence earlier this month, but Fallin said it was not "credible evidence" of his innocence and refused to delay his execution.
Supporters argue that there is no physical evidence to link Glossip to the brutal murder and that Sneed had every motive to lie.
However, Oklahoma Attorney General Scott Pruitt has said he was confident that the criminal appeals court would conclude that there was "nothing worthy" that would lead it to overturn the guilty verdict.
Glossip's case has also sparked interest for his failed bid to ban a controversial drug used in lethal injections. The Supreme Court upheld the use of the drug midazolam in June, saying it does not violate the US Constitution.
Critics, however, say midazolam is not strong enough to prevent inmates from experiencing agonizing pain during executions.