Taoiseach Enda Kenny has said he had not given any directive to people in his party not to vote for John McNulty and he was not going to speculate about the outcome of the Seanad by-election.
Minister Charlie Flanagan denied there was anything untoward in Fine Gael officials retaining a number of votes cast and not passing them on to the returning officer.
Fine Gael said it had done so while trying to clarify the situation and had now forwarded the ballot papers.
It has also emerged that the second IMMA member nominated by Minister for Arts Heather Humphreys was also a Fine Gael member.
The minister has said she was not aware of this at the time.
Fianna Fáil leader Micheál Martin has said Minister Humphreys needs to be clear and name the officials who suggested Mr McNulty's name to her, leading to him being appointed to the board of IMMA.
Speaking on RTÉ's News at One, he said Ms Humphreys and the Taoiseach are continuing to "pedal the line" in saying that it is an internal Fine Gael matter.
He said Mr McNulty's appointment to IMMA did not have anything to do with his credentials for the role.
"Let's not buy into any argument that this was done for reasons to do with the board.
"The minister said he felt he was uniquely qualified and that he had various qualities, but she knew he wasn't going to last two weeks on the board because of the Seanad election."
Mr Martin added that the appointment "clearly wasn't in the interest of IMMA" and he believed the issue "probably reflects a broader neglect of the arts by this Government over the last two to three years".
It has also emerged that ballots cast by Fine Gael TDs and senators in the Seanad by-election last Tuesday were retained by the party.
Under Seanad by-election rules ballot papers must be returned by registered post for counting.
The party said it was holding the ballot papers to investigate whether it was possible for people to vote again if they wished to.
It said the ballots were posted today.
Around a dozen Fine Gael TDs and senators cast their ballots in a party room in Leinster House on Tuesday.
The ballot papers were then collected by a Fine Gael official who undertook to post them.
Later in the day, after Mr McNulty announced that he no longer wished to be considered for the seat, a party official contacted TDs and senators by text to tell them that he would "hold off on posting the ballot papers until the situation was clarified".
When contacted last night, a Fine Gael spokesman said the ballot papers would be posted by registered post today.
He said the ballot papers were retained because the party wanted to investigate whether it was possible for people to vote again for another candidate if they wanted to, rather than a candidate who is no longer in the race.
The spokesman said the party only received clarification last night that once a ballot paper is issued, a duplicate cannot be reissued unless the original is declared lost.
He said there was no intention on the part of Fine Gael to interfere with the democratic process.