Taoiseach Leo Varadkar has told his party colleagues that Cabinet confidentiality should always be respected.
At a meeting of Fine Gael TDs and senators in Leinster House, it is understood that Mr Varadkar expressed deep concern over a leak from Tuesday's Cabinet discussions about Health Service Executive Director General Tony O'Brien.
It was reported after that meeting that ministers Michael Ring, Finian McGrath and Katherine Zappone wanted the HSE boss to step down.
It is believed that the TDs and senators were told that this placed the Taoiseach and Minister for Health Simon Harris in a difficult position.
This is because they were about to go into the Dáil to defend the decision to allow Mr O'Brien to complete his last few weeks in the job.
Mr Varadkar got the backing of former finance minister Michael Noonan, who told the meeting that leaking from a party meeting was disloyal but Cabinet leaks were a criminal offence.
It is understood this view was shared by Minister for Justice Charlie Flanagan.
The meeting also addressed the wider issue of accountability in the health service.
Some were of the view that there was a lack of accountability in the service.
It was agreed the matter would be discussed in detail at a later meeting.