Fine Gael and Fianna Fáil met for four hours Tuesday with further meetings planned Wednesday.
The negotiating teams for the two parties met twice on Tuesday to discuss the viability of supporting a minority government.
Papers were exchanged between the two parties with a paper to be devised Tuesday evening ahead of the next meeting on Wednesday.
Acting Taoiseach Enda Kenny met with his Fine Gael Ministers after the Fianna Fáil talks.
Members of his negotiating team were also holding a meeting with the Independent Alliance on Tuesday evening.
Fine Gael is said to be encouraging the Independent TDs to make their positions known ahead of the Dáil vote on Thursday to elect a taoiseach.
Mr Kenny said on Monday it was his “earnest hope” that a government could be formed.
Within Fianna Fáil there are differing views, although it is acknowledged that a Fine Gael-led minority government is the more likely option to emerge from the latest manoeuvres.