Ruben Gutierrez, a Texas death row inmate, says that newly uncovered DNA evidence proves that he did not stab an 85-year-old woman to death inside her mobile home more two decades ago.

Texas prosecutors argue the testing, whatever it reveals, will not make a difference in his case. On Monday, the Supreme Court heard arguments over whether Mr. Gutierrez could request the testing, appearing narrowly divided on the question.

In July, the court took the extraordinary step of delaying Mr. Gutierrez’s execution just 20 minutes before he was to be put to death. It was the second time the court had halted his execution over legal disputes in the case.

Mr. Gutierrez was convicted and sentenced to death in 1999 for the robbery and the murder of 85-year-old Escolastica Harrison. Prosecutors accused him and two other men of planning to rob Ms. Harrison by luring her from her mobile home and of stealing cash that she kept inside it. Evidence at the trial showed that Ms. Harrison had been stabbed to death in her home.

Mr. Gutierrez maintains that he did not go inside Ms. Harrison’s home and was not aware of any plan to harm her. For more than a dozen years, Mr. Gutierrez has sought to have evidence from the crime scene, including a bloodstained shirt, hair and nail scrapings, tested under the Texas Code of Criminal Procedure, arguing that none of these items would show his DNA.

The argument before the court was not about innocence — Mr. Gutierrez does not contend he played no role in the events that led to the killing — but rather about whether he should face the death penalty.