Update 9.11pm: Former Fine Gael leader Alan Dukes has said he expects there to be a change in the leadership of the party.
The Fine Gael Parliamentary Party will meet next Wednesday to discuss the issue, with Enda Kenny saying he will not comment on the matter until then.
“I think numbers of backbench TDs have lost great opportunities to keep their mouths shut,” he said today.
“I don’t think doing this in the full glare of public speculation is a good way of going about it.”
He added: “My take on it would be that there’s going to be a leadership change in Fine Gael. Given all the circumstances, I think the sooner it happens, the better.”
Update 6.13pm: Simon Harris has tweeted to say that he is working as hard as he possibly can at his job as Health Minister, and has “no intention of engaging in idle speculation”.
Working as hard as I possibly can in my job as Health Minister. Lots to do! No intention of engaging in idle speculation— Simon Harris TD (@SimonHarrisTD) February 19, 2017
It follows earlier reports of what purports to be a leaked Fine Gael Whatsapp message, which appeared to suggest Minister Harris may also be in the running to replace Enda Kenny.
The Whatsapp conversation, which appeared in the Irish Times, showed a message attributed to "Charlie Flanagan" appearing to show him stating: “He hasn’t announced yet but Frances is encouraging him. He wants out of Health”.
Update 3.48pm: A spokesperson for Leo Varadkar has said that he agrees with the timeline for Enda Kenny's departure set out by Housing Minister Simon Coveney today, RTÉ News has reported.
Minister Coveney has said Mr Kenny should step down as Taoiseach after his visit to the White House for the St Patrick's Day celebrations next month.
Fine Gael backbencher Alan Farrell has also broken ranks to demand Mr Kenny stand down as leader, claiming his position was untenable.
One other Cabinet member who has been mentioned as a potential leadership candidate, Minister for Public Expenditure and Reform Paschal Donohoe, ruled himself out of any contest.
Enda Kenny's spokesperson: The Taoiseach will address the [FG] PP meeting on Wednesday and won't be making any comment until then #iestaff— Fiachra Ó Cionnaith (@Ocionnaith) February 19, 2017
Bookmaker Paddy Power has suspended betting on the next Fine Gael leader following a string of large wagers on Mr Varadkar.
Spokesman Feilim Mac An Iomaire said: "If money talks, and it usually does, it's screaming at us that Varadkar is in pole position.
"Betters were willing to back him at any price and we've been left with no choice but to shut the market."
Well that didn't last long! We've now suspended betting on the next FG leader after a string of chunky bets on Varadkar this morning.— Paddy Power Politics (@pppolitics) February 19, 2017
Update 1pm: Fine Gael leadership contender Simon Coveney has said Enda Kenny should step down as Taoiseach once he has travelled to the White House for the St Patrick's Day celebrations next month, writes Fiachra O Cionnaith, Political Correspondent.
The Housing Minister made the remark despite insisting "we don't have a leadership contest that has kicked off yet" and taking a thinly-veiled swipe at Social Protection Leo Varadkar for pushing for Mr Kenny to resign quickly.
Speaking on RTE's The Week in Politics programme, Mr Coveney said he would like to see an "orderly, but quite quick transition of leadership within Fine Gael" once next month's meeting with US president Donald Trump takes place.
Mr Coveney said he expected Mr Kenny to outline his own future in the coming days, but added "that's a matter for him" and that he will not support a motion of no confidence should one be tabled this week.
In a clear reference to his closest competitor Mr Varadkar, Mr Coveney added that he is not going to "set out timetables as others have done", a comment linked to Mr Varadkar's weekend statement calling on Mr Kenny to "settle" the "distracting" issue.
While both Mr Coveney and Mr Varadkar are effectively sending out the same message about the need for Mr Kenny to stand aside, both have taken markedly different approaches in how the message is being given.
The differing strategies are believed to be because supporters of Mr Varadkar want their candidate to show leadership by moving quickly, while Mr Coveney's supporters are keen to target a large cohort of Fine Gael members who feel Mr Kenny should be shown respect and allowed to leave with dignity.
Update 12.46pm: A former leader of Fine Gael says Leo Varadkar has 'played the game badly'.
Pressure is mounting on Enda Kenny to announce when he will step down as leader, in the wake of the garda whistleblower controversy.
Last night, one of the main contenders to succeed the Taoiseach, Minister Varadkar, called the current situation distracting and destabilising for the Government.
Former leader Alan Dukes says a number of TD's have gone about things in the wrong way.
"Some of them, including possibly Leo, have played the game rather badly in the sense that I don't think that you start making moves in a vacuum.
"You should have things a good bit more organised before you make a move and I don't see evidence, although I am not that close in that part of Fine Gael, I don't see evidence that that's happened. But given that point, where we are now, the sooner that happens the better."
Also a main contender to succeed the Taoiseach, Simon Coveney appeared on RTÉ's This Week in Politics today.
When pressed about his own views versus Leo Varadkar's regarding the McCabe scandal, Coveney said: "He (Varadkar) wants clarity from the Taoiseach on timelines, in my view we should trust the Taoiseach on this."
NEW: Leo Varadkar in politely-worded demand for Enda Kenny to end "destabilising" leadership speculation pic.twitter.com/Kp6U5I7doA— Gavan Reilly (@gavreilly) February 18, 2017
When asked if the Taoiseach should clarify if he is going to depart in four weeks, eight weeks or 12 weeks, he said that, "the Taoiseach has led Fine Gael for 15 years, he has led the country through a storm."
Fine Gael TD's have been making their voices heard this week in regards to the Taoiseach's departure, but Coveney said that this won't continue.
"We're not going to keep going like this... it's never an easy process but I think we have an obligation to the country."
Coveney made it clear that he is supportive of the Taoiseach going to Washington for St Patrick's Day but is adamant that the transition will happen soon after.
"My opinion is that the Taoiseach should go to Washington... but I think after that visit you will see an orderly but quite fast transition after that," he said.
Earlier: Fine Gael MEP Brian Hayes has said it would not be in the country's interests to see Enda Kenny removed from office quickly but hinted the Taoiseach will set out his position later this week.
Speaking on RTÉ's Marian Finucane programme, Mr Hayes suggested Mr Kenny must still be in office when the European Council begins talks to set the terms of the EU negotiations after the triggering of Article 50 by the UK and said after his tenure as party leader ended it would be a good idea to give Mr Kenny a key role in Ireland's Brexit strategy.
The MEP went on to say that 'no good would come' from a no confidence motion in the Taoiseach which TDs are threatening to table next week.