In the absence of any physical evidence against him, Mr. Glossip’s conviction was largely based on Mr. Sneed’s testimony. Mr. Glossip’s lawyers noted that Mr. Sneed changed his story several times before he admitted to the murder, then made a deal to receive life in prison in return for implicating Mr. Glossip.

In a new report commissioned by the defense team and released on Friday, Richard A. Leo, an expert on police interrogations, said that based on transcripts of Mr. Sneed’s questioning and testimony, officers had used techniques that were known to cause false confessions — like telling him he would be the scapegoat for the murder, planting the idea that Mr. Glossip was the mastermind and pressuring Mr. Sneed to say so.

The defense also released on Friday an affidavit from a former drug dealer who said he repeatedly sold methamphetamine to Mr. Sneed at the Best Budget Inn, that Mr. Sneed appeared to be addicted and that he paid for drugs with items he stole from cars and rooms in the motel.

Mr. Glossip’s appeals to the state and federal Supreme Court have been exhausted. His last hope is for Governor Fallin, a Republican, to stay his execution while his lawyers work to persuade a judge, or the state board of pardon and parole, that significant new evidence warrants a new hearing or clemency.

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

“We are seriously racing against time, as you can imagine,” said one of those lawyers, Donald R. Knight, from Colorado. “We’re trying to do work that should have been done by trial lawyers a long time ago.”

But Governor Fallin has rejected calls to intervene.

“His actions directly led to the brutal murder of a husband and a father of seven children,” she said last month in a statement about Mr. Glossip, stressing that he had been convicted in two jury trials and lost multiple appeals. “The state of Oklahoma is prepared to hold him accountable for his crimes and move forward with his scheduled execution.”