The Fine Gael parliamentary party meets on Friday to reflect on the outcome of the general election and to begin the process of electing a new leader. Meanwhile, Sinn Féin's parliamentary party will meet on Thursday to discuss a strategy for the 29th Dáil.
Current Fine Gael rules dictate the leader’s position is decided by a combination of TDs and senators but with a greatly reduced number of Dail deputies and Seanad elections not taking place until late July, Friday’s meeting will consider changing the rules agreed at the party’s ardfheis last February.
However, a change to the rules that would increase the level of grassroots involvement would involve a lengthy process which party sources suggest will take too long.
Friday’s meeting will welcome new deputies and discuss the upcoming Seanad elections.
Following the resignation of party leader Mr Michael Noonan at the weekend, speculation on who will succeed the Limerick man has begun to increase.
As yet, no-one has declared their candidature but Dublin TDs Mr Richard Bruton and Mr Gay Mitchell have been mooted as possible leaders along with Mr Enda Kenny who unsuccessfully challenged for the position in the election that saw Mr Noonan become leader.
And despite saying he is not interested in the position, the man who lost the leadership last year, Mr John Bruton, is still seen by some as a possible candidate.
Meanhwile, Sinn Féin’s parliamentary party are to meet with their House of Commons counterparts and the party's Officer Board on Thursday to discuss approach to the new Dáil.
They will then seek meetings with all of the other parties in Leinster House over the next week.
Party president Mr Gerry Adams said: "All of the issues which confronted people during this election campaign will only be resolved through innovative and committed action. There is a democratic imperative on those of us who want to bring about change to work together to bring this about."