The Fine Gael parliamentary party, comprising TDs, Seanad members and MEPs, are meeting again today in an effort to seek consensus on the appointment of a new leader.
This is the second of two days of meetings the party are holding in a Dublin hotel.
Former leader Mr John Bruton last night said that the party needs to have a new leader in place to put forward for Taoiseach on June 6th.
The party are seeking to do so without the need for a potentially divisive election for the position.
Mr Phil Hogan, the Carlow/Kilkenny TD, is the only person to formally announce his candidacy so far.
Other potential candidates for leadership are thought to include Mr Richard Bruton, Mr Enda Kenny, Mr Gay Mitchell, Mr Simon Coveney and Mr Denis Naughten.
If an election is necessary, a majority of the Parliamentary Party now believes that it should not be held until the new Seanad is elected in late July.
