A growing number of Fine Gael TDs believe Enda Kenny should stand down as party leader within weeks.

Some backbench TDs are considering whether they will table motions on his leadership of the party and the country as early as next week.

However, the consensus is there is unlikely to be an organised heave against the Taoiseach in the short-term, although this is being discussed by a small group of TDs.

“This level of political instability cannot go on, given we are a minority Government,’’ said a Fine Gael TD. “Although the Dáil will be in recess, the summer could throw up further controversies leading to an election in the autumn.’’

Mr Kenny last week saw his authority challenged when Independent Alliance Ministers voted in favour of a Bill on fatal foetal abnormalities despite Fine Gael opposing the legislation. Ministers and TDs were also furious when Mr Kenny appointed Dr James Reilly as deputy leader last week.

“James is a nice man, but he is now a Senator who lost his Dáil seat,’’ said a TD. “He does not have the political skills to help rebuild the organisation.’’

‘Time is up’

Another TD said it was a question of the “face on the poster’’ at the next election. “Enda did a good job, but his time is up,’’ said a Minister of State. “We need a new leader and soon.’’

TDs are also concerned that Fianna Fáil will withdraw support from the Government at a vulnerable time and made significant gains in the subsequent general election.

“We are in danger of losing up to 15 seats,’’ said a backbench TD.

A complicating factor is that the election of a new leader could take some time, given the varied nature of the electoral college. Oireachtas members have 65 per cent of the vote, ordinary members 25 per cent and councillors 10 per cent.

The Irish Times reported on Saturday that the party has contingency plans in place to speed up the succession race in the event that Fianna Fáil withdraws its support for the minority Government.

Minister for Social Protection Leo Varadkar at the weekend said Mr Kenny would be a “lame duck” and that his authority would be damaged if he set out a timetable for when he intends to step down as Fine Gael leader.

Mr Varadkar insisted that he did not believe it was time for Mr Kenny to step aside as “we have only just formed a government”. He said Mr Kenny would decide in his own time when he wants to relinquish leadership.

"He has said he does not intend to lead the party into the next general election and that he will go at a time of his choosing,'' Mr Varadkar told RTÉ's The Week in Politics. "I do not think it would be right to push him."

He said Mr Kenny had been given a mandate by the parliamentary party to form a Government after the election and that “he has a lot of work to do”.

Asked about the appointment of Dr Reilly as deputy leader, Mr Varadkar said it was the party leader’s sole discretion under Fine Gael rules.

“I think it is something that came as a bit of a surprise to a lot of people in the party in general, but it is a decision that Enda Kenny is entitled to make,’’ he added. “He has made it and, therefore, I support it.’’