The Higher Education Minister Simon Harris has said comments made by two Galway Mayo Institute of Technology (GMIT) lecturers were "extremely hurtful".
The governing body of the college has launched an internal investigation, after they were heard discussing the performance of some of their students.
The students were able to hear the staff members discuss their presentations and allocate marks, after an online connection remained live.
Some of the clips include insulting language - such as a staff member asking whether there was 'something wrong' with one student.
Several video clips of the incident have been widely shared across messaging apps.
Minister Harris told Pat Kenny the comments were "entirely inappropriate".
"Anybody who heard that language can only but be disappointed and disgusted by it.
"The words were harsh, in many ways were cruel and very, very hurtful to the students involved."
He said educators have a duty of care to their students, particularly now.
"I think it's really important to say that all of us involved in the education space have a duty of care to students - and that duty of care becomes even more prominent and more important in a time of pandemic."
He said that he welcomed the fact that the GMIT president apologised on behalf of the institution.
In a statement on Monday, Dr Orla Flynn apologised for the data breach which caused "such deep hurt and dismay".
She said: "GMIT is known as a student-centred institute and some of the comments made by our staff do not reflect the values to which we aspire.
"We are taking the breaches of GMIT policies and data protection legislation very seriously.
"Apologies will be issued to the students directly concerned."
Minister Harris also said the governing authority is investigating the incident.
He added that the comments made were extremely hurtful, entirely inappropriate and "absolutely shouldn't have happened".
'Unconscious bias'
While Fine Gael Senator and education spokesperson Aisling Dolan has said unconscious bias training should be rolled out to all staff at third-level institutions.
She earlier told Newstalk Breakfast while some people can jump to conclusions, it has to be different for decision-makers.
"This gives us an insight into attitude and behaviours that are just not acceptable."
"In this sort of situation, where there are people making decisions - grading - in this particular incident there were people involved in making assessments about these students.
"And the way that that was being done really is not acceptable - and it shouldn't be acceptable in any walk of life when it comes to making judgements".
She said any such assessments - such as interviews or grading - "has to be based on an evidence base, so it has to be based on factual content".
"I suppose what it comes down to, when we talk about unconscious bias, is being aware how that impacts on our decisions and then ways that we need to mitigate that".
