Update - 1.45pm: Green Party Deputy Leader, Catherine Martin, says that their future probably lies in "constructive opposition".
She said: "We've been in talks with Social Democrats, Fianna Fáil, Fine Gael, Sinn Féin and constructive opposition is what we are leaning towards.
"But I think there's a duty on all TDs not to be silent but to engage in talks, because what is crucial here is that we get the most stable government."
Update - 11.05am: The acting Finance Minister, Michael Noonan, has confirmed that Fine Gael would like to see Labour join it back in government.
Mr Noonan says it is "an open secret" that Fine Gael would prefer to see its former coalition partner back within the fold.
Speaking ahead of talks between Fine Gael and Fianna Fáil this morning, Mr Noonan warned that a second election was still possible.
He said: "It's an open secret that Fine Gael would like Labour to participate, but Labour are now considering that and there seems to be mixed views.
"The two parties that gained seats in the election have ruled themselves out for forming a government, so it's either a former government with the willing or have an election, so it's one or the other."
Update - 9.35am: Fianna Fáil say they will support a Fine Gael/ Labour Government.
Willie O'Dea, Limerick TD for Fianna Fáil, says his party will not oppose Labour backing Fine Gael in a new minority led administration.
He said: "If Enda Kenny can get up to 58 TDs who are willing to participate in government, we'll facilitate that, we'll allow the government to go in.
"I'm not overly concerned about the identities of the extra people he'll get to bring him up to 58, naturally I would prefer if it were more broadly based than just Fine Gael and Labour."
Fianna Fáil and Fine Gael will resume Government formation negotiations later.
Over the weekend it was reported that Labour is now considering backing Fine Gael in a new minority led administration.
Tony Heffernan, who was Press and Parliamentary Director for the Labour Party from 2000 to 2011, believes the delay in forming a Government is one of the key reasons Labour is now considering going back into power.
Mr Heffernan says it means Joan Burton would avoid a leadership election.
He said: "The situation is that if Labour doesn't go back into government, there has to be a leadership election by August, but if Labour does go into government, Joan Burton remains.
"(In that case) There is no requirement for a leadership contest and Joan would remain as party leader presumably for as long as she wanted to, or unless the Parliamentary Party passed a motion in her or something like that."