In June, the Supreme Court ruled against Mr. Glossip and other death row inmates who had challenged the use of a sedative, midazolam, in executions. They argued that the sedative was not powerful enough to prevent excruciating pain from the lethal drugs that follow it, pointing to executions last year in which inmates writhed, moaned and gasped before dying.

Image Richard Glossip Credit... Oklahoma Department of Corrections, via Associated Press

“We anticipate going back to the United States Supreme Court today or tomorrow,” Mark Henricksen, one of Mr. Glossip’s lawyers, said on Monday.

In 1997, Barry A. Van Treese, a father of seven, was beaten to death with a baseball bat, inside a motel he owned in Oklahoma City. Justin Sneed, a 19-year-old drifter who sometimes did maintenance work at the motel, confessed to the killing, but said that Mr. Glossip, who was the motel manager, had talked him into it.

Mr. Glossip was convicted, based largely on Mr. Sneed’s testimony. As part of a plea deal, Mr. Sneed was sentenced to life in prison.

“We now have witnesses who claim that Mr. Sneed has bragged about setting Mr. Glossip up in order to avoid the death penalty,” Mr. Henricksen said.