Poyet, assistant boss Mauricio Taricco and first-team coach Charlie Oatway were told to stay away from the club in the aftermath of their play-off semi-final defeat by Crystal Palace earlier this month.
Brighton are still making no comment on the events surrounding the trio's suspension but, in a statement on the club's website, chief executive Paul Barber revealed the inquiry is ongoing.
Barber said: "The club's internal processes involving Gus Poyet, Mauricio Taricco and Charlie Oatway continue.
"We are conducting these matters reasonably, in light of all the circumstances. These processes will take as long as is necessary and, as I have explained previously, the club will not be making any public comment on the matter until we are in a position to do so.
"We are aware of some ill-informed and unfounded speculation appearing in various places. It is both unfair and unhelpful to everyone concerned. It should stop immediately to allow the internal processes to continue fairly and then conclude appropriately."