"We're going to hopefully build a strong opposition" - Clare Daly
The Independent Alliance says it is expecting a reply from Taoiseach Enda Kenny early next week about its demands for government.
Four of the alliance's six TDs met this morning with Mr Kenny at his request to discuss their ten-point 'Charter for Change'.
It says the Fine Gael leader will reply in detail to their demands early next week.
Waterford TD John Halligan says the Taoiseach himself also put some offers on the table.
Fine Gael and Fianna Fáil held their first parliamentary party meetings since the election yesterday.
There they thrashed out redline topics that would prevent them from forming a coalition.
The final picture of the 32nd Dáil has Fine Gael on 50 seats, Fianna Fáil on 44, Sinn Féin 23, Labour 7, Anti-Austerity Alliance-People Before Profit 6, Social Democrats 3, Green Party 2 and Independents on 23.
The issue of abolishing Irish Water is expected to be the main sticking point, with the two biggest parties taking opposing stances.
Clare Daly says she has not been approached by either Fine Gael or Fianna Fáil about forming the next government.
The Independent TD says she expects to remain in opposition, but says the two parties fear just that.
Deputy Daly was asked if she has been approached by anyone about supporting a minority government.
Meanwhile, the deadline for nominations for the post of Ceann Comhairle is Wednesday March 9th, with Sinn Féin already putting Caoimhghín Ó Caoláin forward.
Independent TDs are also expected to be crucial components of new Dáil.
Independent Deputy in Kerry South, Michael Healy-Rae, says creating a Minister for Rural Development could win their support.
"All of the issues that are important to TDs - whether they're from Kerry, Clare, Mayo, along the west coast of Ireland - the issues such as broadband, the protection of our post office network; all of the things that are important in particularly in rural areas they would come the umbrella of a Minister for Rural Development", he told Newstalk Breakfast.
"And I think that issues like that would be important to independents", he added.
Meanwhile, Fine Gael says it is prepared to form a new coalition government but not at any cost.
In a statement released following its first parliamentary party meeting since the election, Fine Gael says it is critical the new government is a lasting one.
The statement clarifies the party's commitment to supporting some form of water charge.
The Dáil will meet next Thursday to vote on a new Taoiseach, however it is expected to take at least two months to put a new government in place.
It is now thought that whichever leader gets more votes for Taoiseach next Thursday stands the best chance of taking the job eventually.