McNulty has 50/50 chance of winning
Fine Gael Seanad candidate John McNulty is regarded as still having a 50/50 shot of winning the seat in the Seanad, which he is currently trying to lose.
Up to 70pc of Fine Gael TDs and Senators are believed to still have to cast their vote in the Seanad by-election dominated by Mr McNulty's appointment to the board of the Irish Museum of Modern Art (IMMA).
Mr McNulty has asked Fine Gael members not to vote for him, but Taoiseach Enda Kenny says there is no specific instruction to party TDs and Senators on which way to vote.
Health Minister Leo Varadkar has become the first Cabinet minister to openly send a signal to Fine Gael backbenchers not to vote for Mr McNulty.
The minister said he won't be voting for Mr McNulty because the election of a candidate who has withdrawn from the race "would cause further embarrassment to our politics and parliament".
Mr Varadkar's intervention is viewed as significant.
Government Chief Whip Paul Kehoe has followed suit by appealing to Fine Gael backbenchers not to vote for Mr McNulty.
"I'd ask people to respect John McNulty's request for people not to vote for him as he has pulled out of the race," he said.
Mr Kehoe said he will be voting for former Teachers Union of Ireland president Gerard Craughwell, who was nominated by a combination of Independent and Fianna Fail members of the Oireachtas.
Mr Craughwell is actively canvassing Fine Gael TDs, urging them to vote for him as Mr McNulty has withdrawn from the race.
Votes have to be returned by 12 noon on Friday with the count taking place immediately afterwards. A win for Mr McNulty would be hugely embarrassing for Mr Kenny and create a new dilemma for the Taoiseach.
Despite the growing pressure from the party hierarchy to ensure he doesn't win, Mr McNulty is still viewed as having a 50/50 chance of winning the seat.
A lot of Fine Gael TDs and Senators have yet to vote and are believed to be monitoring developments.
"I think it's too close to call," a party source said.
Meanwhile, it has emerged the chairperson of the Fine Gael Women's Group resigned two years ago, accusing the party of "hypocrisy" over the treatment of women.
Businesswoman Frances Cahill wrote to Mr Kenny and Fine Gael general secretary Tom Curran in 2012 telling them of her resignation from both the Fine Gael Women's Group and the governing National Executive.
Mr Kenny has come under fire for nominating Mr McNulty, rather than a woman for the Seanad seat.