Senior figures in Fine Gael have said that they expect a new Government to be in place by Thursday of this week.
The Defence Minister Simon Coveney is hopeful his party will be able to strike a deal with independents to see Enda Kenny elected Taoiseach in the coming days.
Fine Gael need six more TD's to support them before they can set up a minority led administration.
The Defence Minister has also appealed to the Greens and Social Democrats to be supportive of a new Fine Gael Government.
John Downing is a political correspondent with the Irish Independent
