John Halligan and Sean Canney will be among the TDs appointed as junior ministers in the coming days.
The Independent Alliance TDs have been assured of junior ministries but have not been told their roles.
On the Fine Gael side, speculation is mounting as to whom Taoiseach Enda Kenny chooses from his ranks.
A number of first-time TDs have been tipped for appointment, including Josepha Madigan, the Dublin Rathdown TD who had a role in government negotiations with Independents.
Noel Rock, the youngest party member in the Dáil and who nominated Mr Kenny for taoiseach four times, will be hoping to follow in the footsteps of his predecessor Simon Harris and receive a junior ministry.
Meath East TD Helen McEntee, whose father Shane held the position of minister of state, is also tipped for a position.
“It would be an honour, I wold be pleasantly surprised but I am not expecting it at all,” said Ms McEntee.
She added that she would relish the task of minister for state with responsibility for mental health but would also be interested in a junior role at the Department of Education.
As a Fine Gael negotiator, Galway West’s Sean Kyne was particularity respected by a number of the rural Independents and helped gain their trust during minority government talks.
Eoghan Murphy was also a key player in government formation talks and will be hopeful of receiving a call from Mr Kenny.
Catherine Byrne has acted as chair of the Fine Gael parliamentary party and said she enjoys a “good working relationship” with Mr Kenny. “It’s up to the Taoiseach, I will have no say in that, I am happy in what I am doing. Other people will make that decision and we will cross that bridge when we get there,” said Ms Byrne.
The Dublin South Central TD said she had not heard any rumours that she was due to be made a minister for state.
Wicklow’s Andrew Doyle, Offaly deputy Marcella Corcoran Kennedy, Damien English of Meath West, and Carlow-Kilkenny TD John Paul Phelan are all names being floated for positions.
Mr Halligan said: “I am after being told I am getting one, but I won’t know what until Enda Kenny makes his decision and announces it.”
Meanwhile, on Friday Kevin ‘Boxer’ Moran said he and Mr Canney had tossed a coin to decide which of them would receive a rotating junior minister role first, with Mr Canney winning the toss.
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