Green Party leader Eamon Ryan has said the Irish government has "lost confidence" in Ireland's EU trade commissioner Phil Hogan.
Mr Hogan has faced mounting criticism for attending a golf dinner last week with more than 80 people present.
He also failed to comply with quarantine rules on arrival from Brussels.
Taoiseach Micheál Martin has said Mr Hogan has undermined public confidence in terms of the public health guidelines.
Speaking to the media in Dublin, Mr Martin said one of the big challenges around the commissioner has been the changing narrative from him.
Mr Martin said Mr Hogan was not correct in his assertion that a negative Covid-19 test absolved him from the 14-day period of restricted movement.
Ireland's Minister for Health Stephen Donnelly has called on Mr Hogan to consider his resignation.
Previously, Mr Hogan provided details to the President of the European Commission, Ursula von der Leyen, about his time in Ireland leading up to his attendance at the Oireachtas Golf Society event in County Galway on 19 August.
In a joint statement on Tuesday, Mr Martin, Tánaiste Leo Varadkar and Eamon Ryan said they acknowledged Mr Hogan's recent written and public account of his movements when he travelled to Ireland, but said concerns remained.
They welcomed his apology but some ministers and party leaders now believe Mr Hogan's position is no longer tenable.
Speaking on the Irish radio programme, Eamon Ryan said Mr Hogan breached public health advice by attending the event when there was clear government advice to avoid large congregations.
Mr Ryan said the EU trade commissioner was not free to travel around the country, despite having a negative Covid-19 test, saying that this applies in different cases.
He said the Irish government would respect any decision made by the EU Commission so that "we can all move on" and focus on critical economic and international policy issues.
Earlier on Tuesday, Mr Hogan told the Irish state broadcaster RTÉ that he broke no regulations while in Ireland, was no risk to anybody but made big mistakes and is very embarrassed.
Speaking to RTÉ News, he apologised once again for attending the Oireachtas Golf Society dinner.
Mr Hogan repeatedly argued that his negative test for Covid-19 exempted him from the requirement to restrict movements for 14 days.
However, Ireland's Department of Health has said a person is required to restrict their movements for 14 days if they arrive in Ireland from a country not on the green list.
It said the guidance does not state that a negative Covid-19 test shortens the 14-days requirement.
Mr Hogan's primary residence is in Brussels and he arrived in Ireland on 31 July, travelling to his temporary residence in Kildare.
Ireland's Health Minister Stephen Donnelly told the programme Mr Hogan should "listen to the Taoiseach and Tánaiste" and consider resigning.
"What happened in Galway is an absolute disgrace.
"It is a slap in the face for every man woman and family who has made sacrifices in so many different ways," Mr Connelly said.
"What really annoyed me about Galway is that it eroded people's confidence quite rightly, people were furious and they were right to be furious.
"My message is the virus hasn't gone away. We are at a tipping point in Ireland right now. We must do everything we can to avoid another national lockdown."
Also speaking on Morning Ireland, Irish Social Democrats co-leader Róisín Shortall said she believed Mr Hogan's position is untenable.
His "cavalier attitude and reckless and dangerous behaviour" had led to him losing the confidence of the majority of the Oireachtas.
She said the longer the controversy went on, it brought Ireland into disrepute and reflected poorly on the EU Commission.