Minister for Agriculture and Defence Simon Coveney has said he expects a government will be in place by Thursday.
Mr Coveney told RTE he hoped Fine Gael can agree a deal with independents over the next few days.
The minister also called on other smaller parties to think about the role they could play in power.
A deal has been struck between Fine Gael and Fianna Fail to pave the way for the formation of a minority government after two months of post-election stalemate.
After weeks of negotiations, a draft agreement was finally reached on Friday evening.
The parties have established the framework of a pact to enable a Fine Gael-led administration to govern for the period covering the next three budgets in the Irish parliament.
Three previous attempts in the Dail to elect a Taoiseach in the wake of the election ended in failure.
The future of the last government’s controversial water company Irish Water was a key factor in negotiations between Fine Gael and Fianna Fail.
The deal, the full contents of which have not been made public, requires the endorsement of the respective parties’ parliamentary rank and file.
And even with that backing, Fine Gael will need to strike separate agreements with a range of independents before a workable government can be formed.