Critics say that midazolam, a sedative meant to render a condemned person unconscious before other drugs are used to stop the heart, does not always work. The U.S. Supreme Court then opted not to lift the stay for Davis. The U.S. high court is set to hold oral arguments on the request April 24.
But the state high court's ruling halting the executions remains in effect.
Another inmate, Bruce Ward, was also set to die Monday.
Lawyers for Davis and Ward argue the two did not have adequate access to independent mental health experts at trial.
It takes five votes to get most things done at the court, including imposing or lifting a stay of execution. A decision is expected by June.
The state's lawyers pressed the eighth US Circuit Court of Appeals to allow the executions to begin, saying the inmates are simply stalling so a key lethal injection drug will expire. The two are among eight Arkansas inmates facing possible execution this month amid a flurry of legal actions involving all eight cases. The woman was killed in her home after Davis broke in and shot her with a.44-calibre revolver he found there.
"Now some of you may think that someone like me can not be forgiven for all the bad that I have done in this world, but I am not one of them", Davis wrote in what he believed to be his final statement before his execution was put on hold. He fled west and was arrested in New Mexico. Arkansas contends it has acted legally. "It was coldblooded. It was evil".
Davis says now the whole process would have to start over.
Davis was sentenced to death for the 1990 death of Jane Daniel in Rogers, Arkansas.
Even with the stays in place and questions remaining before a number of courts, executions are still possible Monday night.
"The petitioners had their day in court, the jury spoke, and decades of appeals have occurred", Womack wrote.
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Associate Justice Shawn Womack, writing in dissent, said Davis and Ward "had their day in court" and that the families are "entitled to closure and finality of the law".
Protesters marched outside the governor's mansion Monday.
Rutledge was quick to respond to the Supreme Court's decision.
Arkansas has since run into multiple obstacles in carrying out the executions. However, one of the stipulations is that the company cannot sell the drug to federal and state correctional facilities that engage in capital punishment.
Arkansas had originally scheduled eight inmates to be executed in pairs this month before its supply of midazolam expired.
The European Union on Wednesday urged Hutchinson to commute the death-row inmates' sentences.
The state had initially scheduled eight men to die over the course of 11 days, because its supply of one of the drugs it planned to use in the executions - the sedative midazolam - expires at the end of April. The order by the three judge panel lifts the stay by U.S. District Judge Kristine Baker that was issued over the weekend.
(Stephen B. Thornton/The Arkansas Democrat-Gazette via AP). "I can't put myself in his shoes, but I'm profoundly grateful that I was never confronted with that problem".
"The families have waited far too long to see justice, and I will continue to make that a priority", Arkansas Attorney General Leslie Rutledge said late Monday.
In addition to the Department of Correction, the lawsuit named the department's head, Wendy Kelly, and Arkansas Governor Asa Hutchinson as defendants. "We are under the impression, and under the assumption, that those executions will be carried out as scheduled".
They are scheduled for 7 p.m. on Thursday.