Fianna Fáil and Labour say Tánaiste Frances Fitzgerald's position is becoming increasingly untenable because of her refusal to remove the Garda Commissioner from office.
Labour leader Brendan Howlin said the country needed political leadership on the issue of policing.
He said there should be consequences for Ms Fitzgerald if she does not act to remove Commissioner Noirín O’Sullivan from office.
Speaking at the annual James Connolly commemoration in Dublin, Mr Howlin said: "Her (Ms Fitzgerald's) position will become increasingly untenable if that leadership does not manifest in the coming days."
Financial irregularities
Ms Fitzgerald was unaware of allegations of financial irregularities at Templemore Garda college until October 2016.
The Garda Commissioner had been instructed by her legal adviser to inform her in July 2015, but failed to do so for a further 15 months.
Ms Fitzgerald’s department was also aware of the allegations in October 2015, but did not inform her.
The Tánaiste has failed to criticise the Commissioner for the breakdown in communication.
Fianna Fáil spokesman on justice Jim O’Callaghan said Ms Fitzgerald must remove the Commissioner from office. He claimed the Government was attempting to make the Garda-related issues go away.
Political responsibility
Mr O'Callaghan told RTÉ's The Week in Politics he has "reducing confidence" in Ms Fitzgerald for failing to take political responsibility for the ongoing Garda scandals.
Fianna Fáil has said it did not support the retention of Ms Fitzgerald in her portfolio.
Senior figures have raised the prospect of blocking her re-appointment in the event of a new Fine Gael leader being appointed.
Minister for Social Protection Leo Varadkar and Minister for Housing Simon Coveney have expressed confidence in Ms Fitzgerald.
However, neither have stated if they are willing to retain her as Minister for Justice.
