Robert James Campbell, who is due to die in Texas today, sought a stay on the grounds that he may be subjected to an execution as painful as the one suffered by Clayton Lockett on April 29.
Convicted killer and rapist Lockett died 43 minutes after the start of the lethal injection and appeared to be in significant pain. Lethal injections normally take around 12 minutes.
US states using the death penalty face critical shortages of lethal injection drugs after European firms stopped supplying pentobarbital.
Campbell yesterday requested a New Orleans' court's reversal of a denial of the execution stay he has sought.
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Mr Campbell's "8th Amendment rights can only be protected if he is provided the information required to ensure a humane, non-torturous execution," his attorney Maurie Levin wrote. The eighth amendment bars "cruel and unusual" punishment.
Russell Bucklew, who is due to receive a lethal injection next week in Missouri, has argued that a vascular tumor and circulatory problems could put him at risk for the kind of suffering witnessed in Oklahoma.
"These risks are heightened by the use of a compounded drug, pentobarbital, in the absence of any disclosure about the drug's safety, purity and potency. In fact, the Department of Corrections will not even confirm whether the drug is subject to any laboratory testing whatsoever," she said.