Update 5.55pm: The Housing Minister has said he would not like to see the process of electing a new party leader before St Patrick’s Day.
Simon Coveney is among the lead contenders to replace Enda Kenny, who is expected to clarify his position at the parliamentary party meeting tomorrow.
He wouldn’t be drawn on his own candidacy, saying the contest had not yet begun.
Minister Coveney was also asked was he concerned a split might emerge in the party if a timeline was not given by Mr Kenny.
"I hope not. I think and I hope that the Party will trust the Taoiseach to put in place a process that will then, presumably after St Patrick's Day, put a process in place that will allow for an orderly and well-managed transition.
"Don't forget that we have to have the interests of the country first and foremost in our minds."
Update 3.42pm: Enda Kenny has refused to discuss his future as Fine Gael leader.
The Taoiseach refused to give details of his departure to Cabinet - and also refused to elaborate on his plans in the Dáil.
Opposition leaders have asked for clarity on the Taoiseach's plans, which are due to be revealed at a Fine Gael meeting tomorrow night.
However, Enda Kenny says there is no interruption to the work of Government, and insisted that everyone is focused on their jobs.
Responding to Fianna Fáil leader Micheál Martin during Leader's Questions in the Dáil today, he said: "Let me assure you Deputy Martin that there is no change in the Government.
"The Government is a minority partnership Government, made up of the Fine Gael party, the Independent Alliance and a number of independents, so there is no change.
"You can have opinions one way or the other - obviously the position is that Government are focused on doing the job for the people in the country, as was our remit when we were elected."
Update 1pm: Enda Kenny was asked about concerns about "instability" at the heart of government during this morning's Cabinet meeting, which was described as "cordial" and "workmanlike", writes Juno McEnroe, Irish Examiner Political Correspondent
Ahead of an expected announcement by Mr Kenny on his leadership of Fine Gael tomorrow, members held their meeting after a tough week for the government over the garda whistleblower crisis last week.
Cabinet sources said Mr Kenny seemed in "great form", "workmanlike" and cross examined his ministers over their briefs and matters pending, such as reports.
The meeting began just before 10am and finished at noon.
Micheal Martin criticises Gov ministers for "focussing on Simon/Leo show" instead of addressing chronic policy issues at Leaders Questions.— Fiachra Ó Cionnaith (@Ocionnaith) February 21, 2017
Sources said that the issue of his leadership and the tough week for government were not discussed until the very end of the meeting, when Mr Kenny asked his colleagues was there any other business.
Junior minister Finian McGrath then said he wanted to raise something with the Taoiseach. Sources said ministers all remained silent at the table as the Independent Alliance minister then proceeded to ask about the recent crisis.
The minister said he wanted to raise concerns abut "instability in government in the last five days", said sources.
"How are you going to respond and would you like to respond?" the minister in charge of disabilities told Mr Kenny at the meeting.
There was silence in the room before Mr Kenny smiled and replied to the junior minister: "I will deal with that matter tomorrow."
Ministers are understood to have different opinions on whether Mr Kenny tomorrow will set a date or period when he will resign or whether he will stay on for some time and decide this later.
"He [Mr Kenny] was in great form, there was also a sense though he was tidying up," said a source about the Cabinet meeting.
Sources also said that ministers Leo Varadkar and Simon Coveney, the top two contenders to succeed Mr Kenny, were seen talking amicably after the meeting.
Earlier 12.12pm:Fianna Fáil has warned Fine Gael not to allow the race to replace Enda Kenny to become a "circus", saying there is a "limit" to what the fragile confidence and supply deal between the parties can withstand, writes Fiachra Ó Cionnaith, Political Correspondent.
Fianna Fáil finance spokesperson Michael McGrath issued the warning as he refused to be drawn on whether Mr Kenny should go before EU-UK Brexit negotiations formally begin and if the leadership crisis has undermined the value of next month's meeting with US president Donald Trump.
Speaking to reporters at the launch of a Fianna Fáil tendering system reform Bill at Leinster House, Mr McGrath said his party is concerned the leadership issue may impact on the work of Government unless it is properly addressed.
Warning a number of times the matter must not become a "circus", he said while his party remains committed to "honouring" the confidence and supply deal with Fine Gael, the agreement can only withstand so much.
"Our position as a party remains the same, we are committed to honouring the agreement that we have," he said.
"But obviously that holds provided the Government is coherent and is dealing with the issues that are affecting people in their day to day lives, so there is a limit to the amount of instability that can be withstood.
"We're not going to facilitate a Government that develops into a circus. That is our position.
"We are observers in all of this at the moment. Very interested observers, but as soon as the Taoiseach makes his comments to the Fine Gael parliamentary party we would hope that clarity can be brought to the House as well in the coming days so that everybody knows where we stand," he said.
Asked if he believes the leadership issue has already become a "circus", he admitted the last couple of weeks "have been very difficult for the agreement we have in place".
"We don't want a Government that develops into a circus, and the last couple of weeks have not been good enough, quite frankly," he said.
Saying his party is "not going to start laying down points" on when Mr Kenny should leave, Mr McGrath declined to comment on whether the departure should occur before the Brexit negotiations begin.
Similarly, asked about the upcoming St Patrick's Day White House meeting with US president Donald Trump and whether the leadership issue has made Mr Kenny a lame duck and undermined the value of his visit, Mr McGrath said "it is a matter for Fine Gael to deal with any issue they may have".
Earlier 9am: Simon Harris says his age wouldn't be a barrier to him running for the Fine Gael leadership.
The Health Minister said he still hasn't made a decision on running for the post when Enda Kenny steps aside.
Simon Harris said he is glad the Taoiseach will clarify his position tomorrow, and did not rule out running for the leadership.
"I've made no decision in relation to running or not running but I do always make the point that I think age is immaterial," he said.
"I worked extremely hard in government in a range of positions - as Minister for Health, as Minister of State for the OPW, as Minister of State for the Department of Finance, as a member of the Public Accounts Committee, as a TD, and in a range of other areas before I was involved in politics, including disability advocacy.
"I think people should be judged on their vision for the country and not on an age on a birth cert."
The Taoiseach and his ministers have been arriving at government buildings this morning for a cabinet meeting.
Earlier: The five possible Fine Gael leadership contenders will come face to face at a cabinet meeting this morning.
Speculation has mounted that Enda Kenny will announce the timeline for his departure as Taoiseach at a parliamentary party meeting tomorrow.