Minister for Education Ruairi Quinn has refused to comment on reports that he suggested to last week's Labour Parliamentary Party meeting that James Reilly was not up to his job.
Launching a report on film and media literacy, Mr Quinn repeatedly told reporters he would not breach the confidentiality of the meeting.
He would not confirm or deny accounts that suggested he had agreed with TDs who said Mr Reilly "had to go".
However, he did maintain that he had full confidence in Mr Reilly as Health Minister.
Sinn Féin and Fianna Fáil have called for Mr Reilly to resign from his position in recent weeks.
Sinn Féin's Mary Lou McDonald had said her party did not have any confidence in him, and he was presiding over "shambles after shambles" in the department.
Fianna Fáil moved a motion of no confidence in Mr Reilly before the Dáil in September.