The Superintendents’ Association have also said they will support the Police Federation action.
NIPSA represent a number of civilian police staff within the PSNI.
The decision has been made following an “accumulation” of factors, union official Tracy Godfrey has said.
Following a lengthy meeting of the Policing Board on Thursday, Mr Byrne announced he may appeal a High Court judgment which stated that two junior officers were unlawfully disciplined.
The officers were punished for making an arrest at a Troubles commemoration event in 2021.
PSNI Chief Constable says he won’t resign over High Court judgement
Ms Godfrey said she was hoping to convene an Extraordinary Departmental Committee meeting next Thursday to discuss a motion of no confidence in the Chief Constable.
“We are going to assess if there is demand for it, and if there is, we will call that vote. It is a discussion about everybody’s views and then we’ll put it to a vote,” she told BBC Radio Ulster’s The Nolan Show.
“I think people have come to the end of the road with how the organisation is being treated and they’ve just had enough.
“We are fearful the same thing could happen to police staff if a given situation arose. Would we be suspended, repositioned for something that is clearly not a discipline matter.
“We are usually treated in a different manner to our police officer colleagues in many respects.
“There is a real concern about the leadership of this particular situation and how it has impacted on police officers and the potential impact on police staff.”
As well as the fallout from the Ormeau Road incident, a build-up of recent pressures has caused the union to take action.
“It’s been one disaster after another, an accumulation of factors over the years,” said Ms Godfrey.
“The data breach after being told by many years by DoJ that police staff aren’t under the same kind of threat as police officers when we always have been.
“Our names are on it. We are not recognised as police officers so we have no way of defending ourselves if someone is knocking on our door.
“Also, the fact that we’re just not paid the same amount of danger money as police officers and that has been going on for 33 years. Good will is now gone.
“I despair of what needs to happen for us to actually get an uplift in that.”
Mr Byrne said they would be considering an appeal to the High Court ruling. Pic: Liam McBurney/PA Wire
Ms Godfrey stopped short of calling for the Chief Constable to resign.
“I can’t tell him to resign if that’s his stance,” she said.
“But once we have the various meetings, the SANI, PFNI and ourselves, and if we all come to the same conclusion, that might be a different scenario for him.”
The meeting will also discuss the future of other senior PSNI management, but Ms Godfrey would not be drawn on their identities.
“We do feel from reading through the judgement and the diary entries that it goes beyond the Chief Constable,” she said.
“I’m not prepared to say, that’s a matter for the committee to decide. Not all are down to him [Mr Byrne], but the poor handling of this situation on the Ormeau Road especially, the lack of morale in all staff has now come to a pinnacle.”
DUP call for PSNI Chief Constable Simon Byrne to resign ahead of crunch Policing Board meeting
It comes after Police Federation Chair, Liam Kelly, said that officers were “infuriated and antagonised” by the Chief Constable’s statement on exploring an appeal to the High Court ruling.
“Our members and representatives have already expressed disbelief and anger at the hugely disappointing and unexpected statement from the Chief Constable,” he said.
“This has infuriated and antagonised the rank and file further, and once again, the two officers at the centre of the case are being treated disdainfully. It is hugely damaging to officer morale and confidence and has to be condemned.
“I had hoped to refrain from saying anything publicly until the extraordinary meeting of my Executive Central Committee next Wednesday.
“However, such is the outrage that has been expressed by the rank and file, on behalf of the two officers and my colleagues, I feel I have no other option but to call it out – in short I am disgusted, disillusioned and extremely angry.”
Mr Byrne made a statement confirming that he won’t be quitting as PSNI Chief Constable in the aftermath of a marathon seven-hour emergency Policing Board meeting.
Mr Justice Scoffield said the decision was made to discipline the officers to allay any threat of Sinn Fein abandoning its support for policing.
