Other witnesses have also described statements by Mr. Fennell that the Innocence Project lawyers say warrant further investigation. A former insurance sales representative said he had heard Mr. Fennell say he would kill Ms. Stites if he caught her “messing around.” Charles W. Fletcher, a former friend of the couple, said Mr. Fennell had complained that Ms. Stites was cheating on him. And Jim Clampit, a former sheriff’s deputy, said that at Ms. Stites’ funeral, Mr. Fennell looked at her body and said, “You got what you deserved.”

Mr. Fennell’s lawyer, Robert M. Phillips, said that Mr. Fennell denies killing Ms. Stites, and that the Innocence Project was merely recycling claims that were made at trial. It was inconceivable, he said, that the people now coming forward would have stayed silent for so long if their accounts were true.

Mr. Phillips said Mr. Fennell had converted to Christianity, had found a job and a girlfriend, and was “living a law-abiding life.” He said he did not know whether Mr. Fennell supports the death penalty for Mr. Reed.

Could the governor stop the execution?

Mr. Reed’s lawyers and his supporters have pleaded with Governor Abbott to delay the execution by 30 days and to order the Texas Board of Pardons and Paroles to investigate the possibility of commuting Mr. Reed’s sentence.

Mr. Abbott’s office did not respond to a request for comment on Tuesday. The governor has stopped just one execution in nearly five years in office, while allowing 47 to go forward, according to The Texas Tribune.

The one case came in February 2018, when the governor granted clemency to Thomas Whitaker, who had been sentenced to death for killing his brother and mother. Mr. Whitaker’s father, a survivor of the murders, had asked the governor to spare his son’s life, and Mr. Whitaker had agreed to waive his right to seek parole, meaning he would spend the rest of his life in prison.