Outgoing Ceann Comhairle Sean Ó Fearghaíl has signalled an interest in serving a second term in the role - however Fianna Fáil's precarious Dáil numbers may ultimately rule him out of contention.
Mr Ó Fearghaíl was automatically returned to the Dáil as a Fianna Fáil TD for the constituency of Kildare South last weekend after serving as the chair of the 32nd Dáil for the past four years.
Mr Ó Fearghail was elected via a secret ballot in 2016 under reforms introduced through so-called new politics. His election broke with the tradition of the position of Dáil chairman being occupied by a government TD.
However, Fianna Fáil has just one more seat than Sinn Féin in this Dáil, which may mean the party decides not to put forward a candidate.
Speaking before yesterday's parliamentary party meeting, Mr Ó Fearghaíl said he would not be canvassing for the seven nominations need to contest the election for Ceann Comhairle which will be the first act of the 33rd Dáil when it meets next Thursday.
"If there are seven people willing to nominate me I'd be willing to go forward. I am uncertain given the vagaries of the situation. If I get seven nominees I will have to consider it. I've enjoyed the job, it's a great privilege," Mr Ó Fearghaíl said.
He suggested the current political stalemate required a "steadying force and a safe pair of hands", and added: "No one can predict with any degree of certainty as to what is going to come out of this. Because I am still an office holder I won't be canvassing. I can't be putting myself about. The best I can do is respond to whatever happens."
Outgoing Labour leader Brendan Howlin has been widely mentioned as a possible contender having served as Leas-Cheann Chomhairle between 2007 and 2011. However, Mr Howlin said on Wednesday he had no interest in the role
Senior Fine Gael figures have suggested Bernard Durkan as the party's nominee after he stood for Ceann Comhairle and Leas-Cheann Comhairle in 2016 - but failed to get elected to either role.
"If the party feels I have some role to play I'll consider that," Mr Durkan said. "There are votes to take place. There is a need for serious negotiations to take place to see how the numbers break down. It will be premature for me at this stage before next Thursday to give any indication one way or the other."
Sinn Féin has not yet decided whether it will put forward a candidate for the role. In 2016, the party nominated Caoimhghín Ó Caoláin, who retired from the Dáil at this election.
Independent TD Seán Canney was reported to have canvassed TDs in recent days about a nomination.
However, Mr Canney last night denied he was interested in the role.
Several TDs have privately said it would be preferable, if possible, to nominate a female candidate for the position given that no woman has ever served as Ceann Comhairle.
Independent TD Maureen O'Sullivan, who is now retired, stood for the position in 2016 but was eliminated on the third count.