Rule changes came into force on Thursday allowing case reviews to be opened up to victims and the press as part of a bid to remove the secrecy around the process which determines if inmates should be freed from jail or stay behind bars.
The prisoners in question, Government ministers and officials will also be among those who can request the case is not discussed behind closed doors.
The Parole Board confirmed on Thursday a request for Bronson's case to be heard in public has been received and will now be considered.
It is understood the application was made on his behalf.
The number of hearings where the overhaul of procedures applies may be limited at first as the criteria requires them to remain private unless parole bosses deem it is in the “interest of justice” to open up the sessions.
In deciding whether to grant a public hearing, the board’s chairman could consider factors including the wishes of victims and their families as well as any “undue emotional stress” this could have on prisoners.
A Parole Board spokeswoman said: “The new Parole Board rules make it possible for parole hearings to be held in public for the first time in some cases where it is in the interest of justice to do so.
“It is important to state that the normal position will be for parole hearings to remain in private to ensure that witnesses are able to give their best evidence and that victims are not put in a position that could lead to re-traumatisation.”
There must be a “good reason to justify a departure from the general rule”, the board said.
Armed robber Bronson has spent almost 50 years behind bars after repeatedly having his sentence increased for attacking prison staff.