The first and deputy first ministers used powers under what is known the Royal Prerogative to create the role for former BBC journalist David Gordon.
Mr Gordon edited the BBC's Nolan Show.
The Newsletter has discovered the law was changed using the highly unusual procedure to enable the post to be filled without being advertised.
The Ulster Unionist Party leader Mike Nesbitt, who is the leader of the Opposition at the Northern Ireland Assembly, said he was not made aware of the move.
Jim Allister, leader of the Traditional Unionist Voice (TUV), accused First Minister Arlene Foster and Deputy First Minister Martin McGuinness of "flagrantly breaching due process".
"How many spin doctors does Stormont need? They already have 161," Mr Allister told the Newsletter.
"How ironic that the latest appointee is the very person who hitherto would have railed against and exposed the contrived process by which he was appointed."
When contacted by the BBC, Mr Gordon said he could not comment on the issue.
A spokesperson for the Northern Ireland Executive Office said: "It is normal practice in politics in London and Dublin as well as other devolved governments for ministers to select the people who provide them specialist communications advice.
"Those trying to throw dirt know that very well.
"The appointment of the executive press secretary is in full accordance with the powers at the disposal of the first minister and deputy first minister and they make this Order under the powers conferred on them by section 23 (3) of the Northern Ireland Act 1998."
The office's spokesperson described the reaction from opposition politicians as "panic and hysteria".
Mr Gordon is also a former Belfast Telegraph journalist and following his appointment on Tuesday, the first and deputy first ministers expressed delight.
They said they wanted their new press secretary to provide them with expert advice and spearhead the communication of Stormont's policies.
However, the method used to hire him has raised questions from serving and former public appointments commissioners.
The current public appointments commissioner Judena Leslie has written to the head of the Northern Ireland Civil Service over the appointment.
In a statement on Friday, she said: "It is in the public interest that the principles of openness and transparency are adhered to."
Opposition politicians have expressed concern about the use of a special legal procedure to appoint a new Stormont press secretary.