Primary principals from 24 north Down schools have called for transfer tests to be suspended in 2020.
The group includes heads from Bangor, Holywood, Millisle, Ballywalter, Donaghadee, Carrowdore and Portavogie.
In an open letter they said "we do not think that the normal academic tests should be carried out in these abnormal circumstances".
Five Catholic grammars in Newry and Kilkeel had previously said they would not use the tests for 2021.
Northern Ireland's biggest integrated post-primary, Lagan College in Belfast, has also made the same decision.
In their joint letter, the north Down principals said that they were particularly concerned about the impact of school closures on pupils in primary six.
'Period of isolation'
"From mid-March these pupils have been at home and in a lockdown which is likely to continue into June," they said.
"This unprecedented period of isolation and home-schooling is having, and will have, an unknown impact on every pupil.
"Each pupil's emotional and mental well being is vitally important to us and the added pressure of sitting academic selection tests in November and December is inevitably adding to the stress normally associated with these tests."
They said the debate over whether to hold the transfer tests in November and December should focus on pupils and their best interests.
They also said that the normal teaching that pupils would have had before the tests would be "impossible" as they had been out of school since March and would face a "phased" return in September.
"It is also very likely that many pupils returning from this extended absence will need additional personal and emotional support," they said.
"In addition, with the possibility of a second wave of Covid-19 during the autumn, we do not think that the normal academic tests should be carried out in these abnormal circumstances."
"Primary school teachers and leaders are not in a position to make judgements on which pupils should or should not be admitted to a grammar school.
"We call on those with responsibility for the admission of pupils into Year 8 to review their arrangements for 2020/2021 and to prioritise the needs and well-being of the current Year 6 pupils and their families.
"We also appeal to the principals and boards of governors in post-primary schools which use the selection tests to suspend their use for admission in September 2021."
The letter was signed by principals from 24 primary schools in north Down and the Ards peninsula.
Those primary schools are: Ballyholme, Ballymagee, Ballywalter, Bangor Central Integrated, Bloomfield, Carrowdore, Clandeboye, Crawfordsburn, Donaghadee, Glencraig Integrated, Grange Park, Holywood, Kilcooley, Kilmaine, Kircubbin Integrated, Kirkistown, Millisle, Portavogie, Rathmore, St. Comgalls, St. Malachys, St. Patricks, Towerview, and Victoria Primary in Ballyhalbert.