Tánaiste Joan Burton has said she spoke to Taoiseach Enda Kenny at length this morning about the issue of the appointment of people to State boards.
She said she would like to see positions advertised and that Minister for Public Expenditure and Reform Brendan Howlin will be bringing a memo to Cabinet on this issue.
Ms Burton said it was standard practice for all parties to nominate replacements to the Seanad and she does not see this changing.
She said: "These bodies account for a very significant proportion of activity in Ireland.
"They employ huge numbers of people and they're absolutely critical to the whole economic well-being, and indeed in the case of the cultural institutions, to the cultural well-being of the country."
Mr Kenny said on Friday that he took full responsibility for the controversy.
Meanwhile, one of the candidates in the election, Gerard Craughwell, has said he will be writing to the Clerk of the Seanad and also to the Judicial Referee involved in his original objection to the candidacy as he wants both of them to revisit the matter.
Mr Craughwell issued a statement this evening saying he had sought legal advice and following this advice he now intends to “exhaust all electoral law procedures” before he seeks recourse to the courts.
He said he will again write to the Clerk of the Seanad asking that his original complaint be “re-instigated” and he will ask the Clerk to refer all matters to the judicial referee for consideration.
Earlier Minister for Education Jan O'Sullivan said she will be voting for John McNulty in the upcoming Seanad by-election.
The minister said that there were lessons to be learned from what she said had not been a well-run process.
Speaking on RTÉ's News at One, she said we needed to ensure that public appointment procedures were applied right across the board.
Ms O'Sullivan said she did not know Mr McNulty, but he did seem like somebody who had the qualifications necessary to be a good senator.
Fine Gael has been under pressure after Mr McNulty was appointed to the board of the Irish Museum of Modern Art.
The Opposition said the appointment was made to enhance his candidacy in the Seanad by-election campaign.
Mr McNulty resigned from the IMMA board last week.
However, a spokesperson for the returning officer in the Seanad by-election has confirmed that Mr McNulty's membership on the IMMA board is on the ballot paper.
The spokesperson said the official list of candidates for the by-election was completed last Monday. It was signed in the presence of a judge and that is what is printed on the ballot paper.
Elsewhere, the National Campaign for the Arts has called on its supporters to ask their local TDs and senators not to support Mr McNulty's campaign.
In a statement on its Facebook page, the volunteer-led group said the McNulty appointment to the IMMA board flies in the face of any transparency regarding arts decision-making.