A Fine Gael parliamentary party meeting has heard criticism of the Taoiseach's appointment of James Reilly as deputy leader.
Cork South West backbencher Jim Daly claimed that if there had been a secret ballot on Senator Reilly and the deputy leadership, TDs and senators knew what the outcome would be.
The appointment was also criticised by Carlow-Kilkenny TD John Paul Phelan.
Senator Reilly defended his position as deputy leader and set out what he intended to do.
He also claimed he was being used as a proxy by those unhappy with Mr Kenny's leadership - a point challenged by Mr Daly.
TDs including Louth's Fergus O'Dowd reiterated calls for Mr Kenny to spell out his retirement plans.
It is understood that Kerry TD Brendan Griffin, who had called for the Taoiseach to step down by September, this evening said his issues had not yet been addressed.
The Taoiseach was defended by a number of speakers, including deputy ministers Marcella Corcoran-Kennedy and Catherine Byrne as well as Senator Paul Coghlan.
Many said Minister Byrne gave an emotional speech where she appealed for loyalty to the party or otherwise, she said, people should become an independent.
Mr Kenny told the meeting the Government had a heavy work programme and said the party should settle down.
The party, he said, should support each other and said that a line should be drawn under what has been said.
He offered to meet his critics over a cup of coffee.
Mr Kenny's remarks drew a round of applause from the meeting.
At the end of the meeting, some of those who raised issues such as Deputy O'Dowd, said they were loyal to the party.
Senator Paul Coghlan, Leas Cathaoirleach of the Seanad, said they should "keep their captain".
The issue of Enda Kenny's leadership was raised at last Wednesday's party gathering and was followed by speculation and rumblings throughout the weekend.
This week Mr Griffin said it was his preference to see a change of leadership before the Dáil resumes in the autumn.
However, he did not receive public support from any member of the parliamentary party with most members stating it was a matter for Enda Kenny.
Ross says relations with Fine Gael 'pretty good'
Minister for Transport Shane Ross has said that relations between Fine Gael and the Independent Alliance are now "back on an even keel and pretty good".
Speaking ahead of this morning's Cabinet meeting, Mr Ross said there had been "difficulties" last week but he said "things were going a lot better this week" because there had been "a lot of conversation".
On Sunday, Minister for Social Protection Leo Varadkar said the Alliance had destabilised the Government and damaged their own brand when taking a free vote on the fatal foetal abnormalities bill despite advice from the Attorney General that it was unconstitutional.
Mr Ross this morning said he wanted to ensure that there was a stable Government and that their differences had been "put behind them'".
