Supporters of Alan Kelly believe he will be able to run against Brendan Howlin for the leadership of the Labour Party.
Limerick TD and former minister for education Jan O’Sullivan is under pressure to second Mr Kelly’s nomination and allow a leadership election take place.
Some of Ms O’Sullivan’s local supporters are thought to be keen to facilitate a contest between the two contenders.
The seven TDs met privately on Tuesday to discuss the plans to succeed former tánaiste Joan Burton.
The majority of the parliamentary party had been in favour of one agreed candidate being put forward, with Mr Howlin identified as the preferred choice.
However Mr Kelly, the party’s deputy leader, made a plea to his colleagues at the meeting to allow a contest to take place.
The Tipperary TD said he had been contacted by members and supporters from across the country who wanted to have a vote on who the next leader should be.
Nominees for the role need a proposer and a seconder. Mr Kelly can propose himself but needs a TD to second him.
Thirty party members, including several councillors, wrote to chairman Willie Penrose calling for a contest to take place.
The letter insisted a consensus candidate was “the wrong course of action” and a vote of members should be facilitated.
“This is the most equitable way of ensuring that the membership has a full voice in who is to become the next leader of the party, especially after what was a very difficult and bruising election,” the letter reads.
“We therefore call on you and on the party chairperson to ensure that the members have a vote to decide the new leader, and that should any TD in the PLP [parliamentary party] that wishes to run be facilitated with a nomination and duly seconded where necessary.”
Ms Burton and Mr Penrose have insisted they will not become involved in the succession race. However, Mr Penrose has indicated he would prefer a consensus was reached and one candidate chosen.
The remaining Labour TDs have discussed the leadership several times and are understood to favour Mr Howlin as the agreed candidate.
Cork TD Seán Sherlock said he was contemplating contesting the leadership.
However, he is understood to favour Mr Howlin in the event of a contest.
Labour has five Senators, including former ministers Aodhán Ó Ríordáin, Kevin Humphreys and Ged Nash. However, under party rules, Senators cannot propose or second a nomination for leader or deputy leader.
Nominations for the election of leader and deputy leader opened on Saturday and will close on Friday at noon.
If a contest is required, ballots will be issued to 4,000 members on May 23rd, which must then be returned by June 17th. The party’s executive board has agreed the count will take place immediately.
The party also on Tuesday discussed its future, how it can rebuild, and how it would operate in opposition.