Teina Pora's legal team has filed an application for compensation for wrongful conviction and imprisonment.
Mr Pora spent more than 20 years in prison for the rape and murder of Auckland woman Susan Burdett before his conviction was quashed by the Privy Council in March.
Justice Minister Amy Adams said, as no retrial was ordered, she would seek advice from justice officials on the merits of the application.
She said officials may advise her on appointing a Queen's Counsel or retired judge to determine whether Mr Pora can prove he is innocent on the balance of probabilities, which would be a prerequisite for compensation.
Mr Pora's principal lawyer Jonathan Krebs said no other details about the application would be released to avoid the process playing out in public.
An associate professor at the University of Auckland's law school, Bill Hodge, said factors including the original confession would be difficult to overcome if Mr Pora needed to prove extraordinary circumstances of innocence.
"Any investigator is going to stumble over some difficulties for Teina Pora, one being a form of confession. Now, presumably, because of his state, we now disregard that confession but still that's something that is to be taken into account."
Mr Hodge said the investigation would take some time and he did not expect to see the outcome until next year.