Fine Gael leader Leo Varadkar could be elected taoiseach by a narrower Dáil margin than the vote Enda Kenny achieved last year.
Mr Varadkar is expected to be elected taoiseach when the House resumes on Tuesday, June 13th.
He is set to meet Independent members of the Government on Tuesday night and Wednesday, and Fianna Fáil leader Micheál Martin on Wednesday.
He will need the backing of the Independents and the abstention of Fianna Fáil to be elected taoiseach.
Like Mr Kenny, Mr Varadkar will need the support of at least 58 TDs.
Mr Kenny was elected by 59 votes to 49 last May. Those who voted for Mr Kenny included the 50 Fine Gael TDs, the Independent Alliance – comprising Shane Ross, Finian McGrath, Kevin "Boxer" Moran, Sean Canney and John Halligan – as well as Katherine Zappone and Denis Naughten, the other two Independent members of the Cabinet.
Clare TD Dr Michael Harty also supported Mr Kenny, as did Tipperary's Michael Lowry, although neither are members of the Government.
Another group of Independent TDs – Michael Healy-Rae, Michael Fitzmaurice, Maureen O'Sullivan and Noel Grealish – abstained.
However, Mr Fitzmaurice has said he will vote against Mr Varadkar’s nomination. The Roscommon-Galway TD almost joined the Fine Gael-Independents minority government last year.
He did not do so because of an inability to reach a deal with Fine Gael on the issue of turf cutting, but now he says he abstained because he believed the deal would be delivered upon last year. “All that has gone by the wayside now,” he has said.
Mr Healy-Rae said he had yet decided what to do in the vote for taoiseach. Mr Grealish has not yet decided what he will do either.
Ms O’Sullivan said she too had yet to decide an approach, but added: “I don’t see the point in voting against the person. I’ll probably abstain if not vote for him.”
Mr Lowry is understood to be considering his position, too. Dr Harty, along with Ms Zappone and Mr Naughten, is due to meet Mr Varadkar on Wednesday. The Independent Alliance members will meet Mr Varadkar on Tuesday night at 8pm.
Future of healthcare
Dr Harty said he will ask Mr Varadkar about his views on the recently published report from the Oireachtas Committee on the Future of Healthcare before deciding what to do.
The Clare TD has remained on the Opposition benches and supported the Government on a case-by-case basis. He said he did not expect to be formally asked to join the Government by Mr Varadkar, but did not rule out such a prospect entirely – although he described it as a “hypothetical” scenario, adding that such a “likelihood is minimal”.
Sources in Mr Varadkar’s camp said they were prepared to talk to Dr Harty, but did not give any specifics of what this could entail.
Meanwhile, former Fine Gael minister Lucinda Creighton has said she has "no plans at all" to rejoin the party. Ms Creighton lost the Fine Gael whip in 2013 after voting against the Protection of Life During Pregnancy Bill, and later formed Renua.
Ms Creighton, who lost her Dáil seat last year, and Mr Varadkar were close friends. Former Wicklow TD Billy Timmins, who also helped form Renua after leaving Fine Gael, said he wanted to stand for the Dáil again and added that he "believes in party politics as opposed to the Independent concept".
He said he has "severed ties with Renua" but would not be drawn on whether he wants to rejoin Fine Gael. *Another former Renua TD, Terence Flanagan, is expected to rejoin Fine Gael.
*Article amended on Tuesday, June 6th, 2017