A military judge is weighing whether to resume hearings in the Sept. 11 case despite two potential hitches: Prosecutors are still awaiting word from the Biden administration on a proposed plea agreement, and a court-ordered examination is underway to determine whether one of the defendants, Ramzi bin al-Shibh, is sane enough to face charges.

Hearings in the case have been on hold since prosecutors initiated plea talks a year ago. But in a recent order, the judge, Col. Matthew N. McCall, invited prosecutors and lawyers for the five defendants to propose what issues might be tackled in a hearing in July at Guantánamo Bay.

In April, the judge ordered the appointment of a panel of three mental health experts to investigate whether Mr. bin al-Shibh “is suffering from a mental disease or defect that renders him mentally incompetent to stand trial.” A report is due July 13.

A finding of incompetence would disqualify Mr. bin al-Shibh from pleading guilty or being prosecuted. It could lead to a court inquiry to determine what mental health providers might do to restore his competency, including possibly forcing him to take more aggressive psychotropic drugs.

The evaluation is occurring at a time of outspoken international concern over the capacity of the Pentagon to provide complex health care at the prison in southeast Cuba. Last month, the chief of the Washington office of the International Red Cross issued a rare statement declaring that the 30 detainees’ “physical and mental health needs are growing and becoming increasingly challenging.”

In his order, Colonel McCall granted the doctors access to information about what had happened to Mr. bin al-Shibh during his four years of interrogations in the C.I.A.’s secret “black site” network where prisoners were tortured.

The entire report will not be made public. Prosecutors will be provided with the decision, not the underlying facts. But the medical investigation may offer insights into the long-term effects of the C.I.A.’s use of nudity, sleep deprivation and physical abuse to get the prisoners to disclose Qaeda plots.

Mr. bin al-Shibh claims that he has been tormented by noises and vibrations as part of a yearslong campaign to deprive him of sleep. Court testimony and filings show that he often screams in the night, berates guards, vandalizes his cell camera and disrupts other prisoners’ sleep. Doctors have given him antipsychotic drugs, although opinions range about his condition — from a delusional disorder to psychosis caused by post-traumatic stress disorder, or other mental health maladies.

One member of the mental health panel is Paul Montalbano, a forensic psychologist who evaluated John W. Hinckley Jr., the man who tried to assassinate President Ronald Reagan. Mr. Hinckley was found not guilty by reason of insanity in 1982 and confined to a psychiatric hospital in Washington for three decades. Dr. Montalbano assessed his suitability for release, which took place in phases between 2016 and 2022.

The identities of the two other panel members are not known.

The question of competency emerged behind the scenes even before prosecutors entered into plea negotiations in March 2022 with Khalid Shaikh Mohammed, the accused mastermind of the Sept. 11 attacks; Mr. bin al-Shibh; and the three other defendants. Prosecutors who had been trying to bring the death-penalty case to trial since 2009 proposed the plea talks as they struggled to comply with the judge’s orders to provide defense lawyers with more information about the prisoners’ time with and treatment by the C.I.A.

In exchange for admitting to their roles in the hijackings, which killed nearly 3,000 people in New York, at the Pentagon and in a Pennsylvania field, the defendants want assurances that they would not have to serve their sentences in solitary confinement. The prisoners, who suffer a range of conditions they blame on their torture, also want the Pentagon to agree to set up a civilian-run trauma care program for them.

Prosecutors have presented some of those questions to the Pentagon general counsel as “policy principles,” which require the administration to take a position on whether they can be accommodated.

Prosecutors have asked to call witnesses during the next hearing, including Frank Pellegrino, a retired F.B.I. agent who took part in the interrogation of Mr. Mohammed in 2007 at Guantánamo Bay.