A spokesperson for the Taoiseach has said the possibility of former garda commissioner Nóirín O'Sullivan retiring was flagged with him in the middle of last week.
Leo Varadkar was formally told about the retirement last Sunday, the day Ms O'Sullivan's retirement was announced.
The Cabinet was today briefed about the process of recruiting the new commissioner which will be handled largely by the Policing Authority.
It was the first Cabinet meeting since the retirement.
The recruitment process is expected to take several months to complete.
However, Acting Garda Commissioner Dónall O'Cualáin has confirmed that he will not be applying for the position of Commissioner.
He said: "I am honoured to have been appointed Acting Commissioner. My focus as Acting Commissioner is to ensure that An Garda Síochána continues to protect and support communities, while progressing the significant changes to the organisation under the Modernisation and Renewal Programme."
Measures to deal with the housing crisis were also discussed by the Taoiseach and his ministers.