The Supreme Court in London will rule later on whether the Police Service of Northern Ireland is sufficiently independent to carry out an investigation into the case of the “hooded men”.
The seven justices will also rule on the lawfulness of the PSNI’s decision to discontinue a probe into the 1971 case.
The 14 hooded men were subjected to a series of controversial interrogation techniques when they were interned without trial by the Army.
The techniques included hooding and being put in stress positions, forced to listen to white noise and deprived of sleep, food and water.
The European Court of Human Rights previously ruled that while the men suffered inhumane and degrading treatment, it fell short of torture.
The PSNI took the case to the UK’s highest court having failed in Belfast’s Court of Appeal to overturn a High Court ruling that found the police should revisit its decision to end its investigation into the treatment of the men.
The Supreme Court looked at that issue and also whether the PSNI is sufficiently independent to conduct the legacy probe.
The justices also examined a legal point regarding how far back the European Convention on Human Rights can be retrospectively applied.
The Supreme Court judgment will also rule on another Troubles incident on Wednesday.
The Army’s Military Reaction Force unit has been suspected of involvement in the killing.
The case before the Supreme Court again centred on whether the PSNI was sufficiently independent to investigate the historical killing.
Lord Hodge, Lord Lloyd-Jones, Lord Kitchin, Lord Sales, Lord Hamblen, Lord Leggatt and Lord Burrows heard legal submissions on the cases earlier this year.