A decision on Ireland's nominations for its next European Commissioner is now expected later this week.
The three coalition party leaders discussed the issue last night and again today without reaching agreement.
They will meet sometime later this week to renew those discussions, but a time has not yet been scheduled.
A government source said that it was a delicate process between the three parties.
It is expected that the Government will put forward two people, a man and a woman, as requested by European Commission President Ursula von der Leyen.
Three senior Fine Gael figures have been closely associated with the position - Minister for Foreign Affairs Simon Coveney, European Parliament Vice President Mairead McGuinness and MEP and former minister Frances Fitzgerald.
Speaking on RTÉ's Prime Time, Minister for State Thomas Byrne said there is no delay in putting forward names to the EU and there is "genuine deliberation regarding the appointment".
He said it is a decision for the Government and there is no dispute between the parties.
Fianna Fáil Minister of State Thomas Byrne & Eoin Drea of the EPP think-tank @MartensCentre discuss who our next EU Commissioner could be and whether Ireland has any chance of retaining the trade portfolio#RTEPT | #Golfgate | @downesr | @thomasbyrnetd | @EoinDrea pic.twitter.com/H4NH31UfPe— RTÉ Prime Time (@RTE_PrimeTime) September 1, 2020
The Social Democrats said appointing Ireland's next Commissioner is a national issue that goes beyond a focus on personalities.
Co-leader Róisín Shortall said it was important that the Government complies with Ms von der Leyen's request as it was the least Ireland could do as it sought an economic portfolio.
Ms von der Leyen has also signalled that she may reshuffle portfolios, which would mean Ireland may no longer hold the powerful trade position previously held by Phil Hogan.
Taoiseach Micheál Martin, Tánaiste Leo Varadkar and Green Party leader Eamon Ryan discussed the issue when they met last night.
Political groupings in the European Parliament, such as the Socialists and Democrats and the Greens, have also called for a female candidate from Dublin, an important issue given MEPs must give their assent.
The European Commission has not yet received any name or names from the Irish Government as candidates to replace Mr Hogan, according to a commission spokesperson.
Amid reports that the Government will submit two candidates, the spokesperson reiterated Ms von der Leyen's determination to ensure gender balance within the College of Commissioners.
Spokesperson Dina Spinant said: "The matter of gender balance is very important to President von der Leyen.
"We have been saying this and she has been saying this very clearly and very loudly since she took office, and actually before taking office, since she was voted in by the European Parliament last July.
"That continues being a matter she pursues with determination. Therefore, she has made the same request to the Irish Government as she has to all governments last summer, to propose two names, including the name of a woman as candidates for the post of commissioners."
Ms Spinant said President von der Leyen would decide on the successful nominee's portfolio once she had "the name on the table".
The portfolio would depend "on the competences and the elements she has at her disposal at that moment to make the decision regarding allocation of portfolios in the College of Commissioners," Ms Spinant said.
Additional reporting: Paul Cunningham and Tony Connelly