The INTO say that there have been some delays in the process since schools reopened, with general secretary, John Boyle saying any delay can lead to an increase in anxiety.
"We don't have the evidence that the testing is as fast as it ought to be and it seems to be confined to situations where there is a confirmed case.
"We have had evidence of some people having to wait three or fours days for the results and the anxiety begins to rise," said Mr Boyle.
"Despite the fact that maybe the only person that knows that there is somebody symptomatic in the school, the principal, in small communities the word get out and people are anxious.
"We are hoping that the testing and tracing will extend to anybody who is symptomatic in the school very quickly," Mr Boyle added.
Earlier this week, schools in Dublin and Kerry sent home groups of pupils after students tests positive for the virus, while a school in Co Clare was forced to close after a number of staff members were identified as close contacts of a confirmed case.
The HSE have confirmed that two pop-up testing centres will be opened, one in Croke Park and one in Castleknock Health Centre to aid testing for the virus.
The two centres, which will open tomorrow, will increase daily capacity by between 180 and 200 and testing will be free for people displaying symptoms who have been referred by a GP.
