Minister for Public Expenditure Brendan Howlin is seeking to run uncontested for the Labour leadership but may face competition from its current deputy leader Alan Kelly. Mr Howlin has spoken to a number of parliamentary party members in an effort to secure the backing to replace Tánaiste Joan Burton, who has come under internal pressure since the party lost a significant number of seats in the general election.

Mr Kelly, the Minister for the Environment, is said to have made contact with Ms Burton on Thursday morning, the day after party members met for a post-election postmortem. He is said to be assessing the level of support among colleagues for his candidacy before deciding whether or not to seek the position. It is understood Mr Kelly feels there is a need for a new generation of Labour politicians to take the reins of the party.

Under Labour’s rules, TDs seeking the leadership have to be both proposed and seconded from within the parliamentary party before the wider party members votes to select the new leader. Party sources believe Mr Kelly would struggle to garner the necessary support within the party to stand.

Mr Howlin, meanwhile, held a meeting with senior Labour official Derek McDowell and his special adviser Ronan O’Brien this week. He has also spoken to a number of Labour TDs to assess if the party can avoid a leadership contest.

Mr Howlin is said to have the support of the majority of Labour members and has secured the backing of Minister of State Seán Sherlock, who was being tipped as a leadership contender from the party’s younger generation.

A senior Labour source said Mr Howlin had made it clear that he would only seek the role if he faced no competition. “If Alan decides to stand Brendan will not,” the source said. “However, Seán [SHERLOCK)]will. He is supporting Brendan but if it is a contest Seán would seek to stand for the position.”

The party met this week to discuss the results of the general election and the performance of Ms Burton, who replaced recently retired TD Eamon Gilmore as leader in 2014. Several of the TDs who lost their seats in February’s election called for Ms Burton to step aside and encouraged Mr Howlin to take up the position.

The party also discussed whether there was a need for a deputy leader in a smaller parliamentary party. Labour won 37 seats in the 2011 general election but just seven this year.

Ms Burton has said she will act in the best interests of the party after hearing the calls for her immediate resignation. She told the meeting she did not believe a government would be formed for another month. The Tánaiste said that after that point, she would do what was in the “best interests of the Labour Party”.

However, the early indications show Ms Burton may face a heave sooner than she expected. Labour's national executive met on Thursday night to discuss housekeeping issues but the issue of leadership was not discussed. Meanwhile, Fine Gael is hopeful Labour may support acting Taoiseach Enda Kenny's nomination for taoiseach this week.

Labour's parliamentary party has voted against entering government but it could still support Mr Kenny in his nomination next Wednesday. Party members had called for Labour to abstain but it is understood Mr Kenny has made contact with Ms Burton to try secure her support.

* An earlier version of this article said the meeting Mr Howlin had held was attended by Pat Montague. Mr Montague was not at the meeting.