Taoiseach Enda Kenny is resisting calls to step down as Fine Gael leader during the Dáil's summer recess.
Mr Kenny told an event in Castlebar this afternoon that he would "not be diverted" from the work he was mandated to do.
Earlier today, Kerry backbencher Brendan Griffin called for a new leader to be elected by September, citing the need to be prepared for a possible general election.
It led to a series of Fine Gael ministers voicing their support for Enda Kenny.
Minister for Finance Michael Noonan said Mr Kenny is a leader Ireland cannot afford to lose just yet.
"I think the country and the party are fortunate to have Enda Kenny as Taoiseach now," he argued.
"He led the government which took the country out of the biggest crisis it had in two generations, when we nearly slipped into bankruptcy.
"He saw the back of the Troika, and he's presiding over a country that's growing at the fastest rate of the 28 countries in Europe."
Minister for Education Richard Bruton said he thought it "a mistake" to raise questions over the Taoiseach's leadership.
Mr Bruton, who led a failed heave against his party colleague in 2010, told Newstalk Breakfast that calls for a new Fine Gael leader were misjudged.
“I believe very strongly that now is not the time to challenge the leadership of Enda Kenny. It is not in the country’s interests,” he said.
“Fine Gael has just taken on the responsibility of confronting huge challenges in housing, health and education. We are trying to make life better for people.
“We have just had the body blow of the British decision to exit the EU. We need Enda Kenny’s experience and his relationships in Europe to steer us through this very difficult time.
“I think it is a mistake to be raise this leadership issue," the minister added.
Call for 'change of leader' before September
The ministerial comments came following reports that Mr Kenny may face a motion of no confidence from within his own party.
In a statement this morning, TD Brendan Griffin called for a new leader to be installed over the course of the next two months.
"A change of leader before the Dáil returns from summer recess in September may be Fine Gael's last chance of an orderly transition to new leadership before the next general election," he added.
Fine Gael "would be decimated" and Fianna Fáil "would most likely return to power in large numbers” if his party chose Mr Kenny’s successor during an election campaign, Mr Griffin added.
He said he would have "major fears for the progress we have made as a country in recent years in that event".
Full statement by @BGriffinTD calling on Enda Kenny to step down before September pic.twitter.com/NoxSOwDWqE
— Catherine Healy (@Chealy_) July 11, 2016
Meanwhile, Fine Gael TD Pat Deering has encouraged his party leader to clarify his position at a parliamentary party meeting on Wednesday.
Mr Kenny previously said he would not lead Fine Gael into the next election. However, Mr Deering told Newstalk that it was important to give his successor time to settle into the role.
“There are a number of outstanding people in Fine Gael who would be more than capable of being leader,” he said.
The Irish Times reports that Fine Gael TDs Jim Daly, Fergus O’Dowd, Fergus O’Dowd and Michael D'Arcy also want the leadership issue to be “clarified relatively soon” after the October budget.
It comes after government chief whip Regina Doherty called on Mr Kenny to spell out how long he will stay on as leader.
"I think it would be fair if [he] would put clarity on that so we could get back to normal," she said on Friday.
But Mr Kenny has the support of other senior party figures, including Minister for Social Protection Leo Varadkar, who was tipped in a poll last week as the voters' favourite to replace him.
Mr Varadkar described Ms Doherty's remarks as "unhelpful" and called on his colleagues to focus on getting "down to business".
Minister for Foreign Affairs Charlie Flanagan also defended the Fine Gael leader this morning, citing his "patience, energy and work ethic".
