The DUP has submitted a motion of no confidence in the PSNI chief constable for consideration by the Northern Ireland Policing Board.
The board is due to have a public meeting next Thursday which Simon Byrne is expected to attend.
The rules state that ten days should elapse between the submission of a no-confidence motion and its consideration.
But there is scope to take it earlier if the board agrees.
There is no guarantee that the motion will get the support of the majority of the 19-member oversight body.
Mr Byrne is under increasing pressure to step down after a court found he had unlawfully disciplined two probationary officers after pressure from Sinn Féin.
Unionist leaders have accused him of bowing to political interference and called on him to quit.
Last night, the chief constable told an emergency meeting of the board that he would not be resigning.
DUP leader Jeffrey Donaldson said that a planned meeting with Mr Byrne had been cancelled this afternoon.
In a statement, he added: "Confidence in the chief constable has been eroded, both amongst the wider public but significantly also amongst serving PSNI officers and staff.
"In light of that we believe that a change of leadership is required.
"Allowing the issue to drift will only cause greater problems for public confidence and for the PSNI as an organisation."
Under the legislation the board can seek the resignation of the chief constable in the interests of "efficiency or effectiveness", but it needs the support of the secretary of state for Northern Ireland.
PSNI staff associations have also expressed anger at Mr Byrne's plan to consider an appeal of the court judgment.
Several are planning to consider motions of no-confidence in him next week.
On Tuesday, the embattled chief constable is due before a Westminster committee in London to answer questions about a huge data breach earlier this month.
It saw the PSNI inadvertently publish details of all 10,000 of its staff.
These included the surname, initials, rank/grade, role, service number, department, location, duty type and gender of all serving officers and civilian staff.
Earlier this week, a judge ruled that two junior officers had been disciplined unlawfully and Mr Byrne had been swayed by suggestions that Sinn Féin would withdraw from Northern Ireland's policing structures if the matter was not dealt with promptly.
Sinn Féin has denied making any such threat.
Having accepted the judgment several days ago, Mr Byrne last night said that he was considering an appeal after seven hours of talks at the Policing Board.
That outraged the union representing the PSNI's rank and file - the Police Federation - which said it was "hugely disappointing".
The federation is to hold an extraordinary meeting next Wednesday at which a motion of no confidence in the chief constable could be tabled.
"In effect he would be appealing against his own officers, dragging this matter out internally and externally," said Police Federation chairman Liam Kelly.
"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."
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The federation represents officers from constable to that of chief inspector, but more senior officers are also backing its stance.
In a social media post, the Superintendents' Association of Northern Ireland, which represents many middle managers in the service, said it was standing with the federation.
It said the "senior executive team must listen to our collective concerns".
Meanwhile, a trade union which represents PSNI civilian staff said it is also considering a confidence vote in the chief constable.
NIPSA is set to hold an extraordinary departmental committee meeting next week, during which it will be assessed if there is a demand for a confidence vote in Mr Byrne.
NIPSA official Tracy Godfrey said she feels it is likely that police staff colleagues will want to have a vote in the chief constable's future.
We stand with our colleagues in @PoliceFedNI & the senior executive team must listen to our collective concerns. We have the best interests of the service at heart. @PoliceServiceNI
— SANI (@SupersAssocNI) September 1, 2023
The DUP has already called on Mr Byrne to resign and the Ulster Unionist Party added its voice to that last night.
UUP leader Doug Beattie said that the chief constable, as well as his deputy, who was also implicated in the critical court judgment, should step down.
He also called for a fundamental review of the Policing Board oversight arrangements, adding that the two probationary constables had been "hung out to dry".
Mr Beattie said the two top police officers should quit "for the good of the service and to enable controlled change in the senior leadership positions within the PSNI".
He said all the facts surrounding how the chief constable had come to the decision he had around the disciplinary process, and who had briefed him on Sinn Féin’s attitude, needed to be properly reviewed.
Additional reporting PA
