A fractured Supreme Court struggled on Wednesday to decide whether to grant a new trial to Richard Glossip, a death row inmate in Oklahoma whose challenge to his conviction gave rise to an extraordinary concession from the state’s attorney general.

Some justices wondered whether they had the power to rule for Mr. Glossip at all in light of a state court ruling against him that was grounded in part in state law. The likeliest outcome, proposed by Justice Ketanji Brown Jackson, was a half-measure that would satisfy neither side: sending the case back to Oklahoma for an evidentiary hearing on the meaning of a prosecutor’s long-suppressed notes.

Mr. Glossip was convicted of arranging the death of his employer, the owner of a motel in Oklahoma City. Two independent investigations cast doubt on his guilt, saying critical evidence had been withheld and major testimony was faulty. State lawmakers from both political parties, along with celebrities like Kim Kardashian, have called for clemency or a new trial.

Paul D. Clement, representing Oklahoma, told the justices that Mr. Glossip’s capital conviction was tainted by prosecutorial misconduct, making a new trial imperative.

“Our prosecutors elicited perjury here, and a man’s going to go to his death,” he said, characterizing the reluctant conclusion of Gentner Drummond, the state’s attorney general, a Republican. “We can’t allow that to happen.”

Mr. Glossip was convicted based almost entirely on the testimony of the state’s star witness, a handyman named Justin Sneed who had pleaded guilty to killing Barry Van Treese, the motel owner, beating him to death in 1997 with a baseball bat.