Fine Gael is to open talks on the formation of a new government, with Labour its likely coalition partner.
However, one frontbencher said this evening that the party was still open to dialogue with like minded independents.
With Fine Gael firmly in the driving seat, the party is trying to highlight Enda Kenny's status as Taoiseach in waiting - a statement tonight listed the European leaders who had congratulated him.
They included Angela Merkel, Jose Manuel Barroso and David Cameron - who has invited Mr Kenny to Downing Street.
Before that the party has to conclude Coalition negotiations.
Labour is the likely partner - but Leo Varadkar has insisted the party is also open to talking to like minded independents.
But Pat Rabbitte told RTÉ News it would be difficult for Fine Gael to enter talks with our European partners on our economic problems if the government it led was relying on independents.
Any talks with Labour would have to conclude by the weekend to allow the party have a special conference.
Counting is continuing in the four remaining constituencies in the General Election, with 12 seats in the new Dáil still to be filled.
Wicklow, where a recount is in progress, has not yet elected any candidates. The Wicklow recount - which was requested by Dick Roche - has been adjourned until tomorrow.
During yesterday's counts, Fianna Fáil saw its first preference vote more than halved, down to just 17.4%, and with a lack of transfers, the party was quickly in serious trouble.
There are now no Fianna Fáil TDs in Meath, Tipperary, Sligo, Kerry, Leitrim or Roscommon.
Even worse news for the Greens, as all six of the party's TDs lost their seats.
Sinn Féin has done well, winning seats it did not really expect to, such as Cork East and Sligo North Leitrim.
The same is true of Labour, taking seats in places such as Clare and Cork South West that had seemed beyond its reach, as well as second seats in Dublin three-seaters North West and North East.
