Parents fear making kids sit transfer tests in grammar school halls could put their health in danger.
Naomi McBurney began a campaign for students to be allowed to take the exam at their own primary schools before Covid-19 hit.
But she said the pandemic, and this week's new coronavirus restrictions, made it unsafe for children to "burst their bubbles" and sit the exam in large groups.
Last week the Health Minister announced new limits on social gatherings.
He also said he expects there will be Covid cases in schools as they reopen this week.
Mrs McBurney, whose petition for pupils to sit post-primary transfer tests in their own school has topped 6,500 signatures since it launched in January, said she also had the support of primary school principals.
She added: "This is imperative now - we’ve only a few months left and it’s [the pandemic] getting worse not better.
"These private providers are saying it’s ok to bring hundreds, thousands of primary school students together from all across NI who wouldn't normally mix together.
"It's large clusters of children coming together that shouldn’t."
But one of the test providers, The Association for Quality Education (AQE), said it would be impossible for kids to take the test in their own school - unless it was allowed at every campus in NI, only there was "no prospect of this happening".
AQE's joint CEO Darrin Barr said each testing centre would have risk assessments.
He added: "The safety of the pupils, staff and parents is AQE's first priority and we will ensure that all of the health and safety guidance, that is relevant at the time of the assessment, is followed.
"It is the opinion of the Board of Directors of AQE Limited that ... hosting the Common Entrance Assessment in primary schools could not be done on an equitable basis unless all schools, who have pupils registered for the assessments, host the papers.
"Since there is no prospect of this happening, the Board has concluded that it will not be possible to host the CEA in primary schools."
A Department of Education spokesman said it was for individual schools to organise with the testing organisations.
He added: "There is no legal impediment preventing primary schools from hosting the transfer tests. "Schools hosting the tests will be responsible for following appropriate public health advice and ensuring that the environment in which the tests take place is safe for young people."
In a statement on its website, test provider Post Primary Transfer Consortium, said it "recognises that its plans may be affected by the unfolding situation".
