Education Minister Peter Weir has rejected a call from the head of the Catholic Church in Ireland for Catholic grammar schools to "suspend" academic selection in 2020.
Writing as Archbishop of Armagh, Eamon Martin said it would be in the "best interests" of children.
He urged schools in his diocese not to use selection to admit pupils in September 2021.
Mr Weir said there is no "viable alternative to put in its place".
The minister told BBC Radio Ulster's Evening Extra programme he acknowledged there were "very real concerns out there", but added: "If we are going to have academic selection then we need some form of a testing system".
The archbishop said testing in the current climate was "cruel", but he acknowledged the issue was "sensitive and complex".
In his role as apostolic administrator for the Diocese of Dromore, he has written a letter making the same appeal to Catholic schools in that diocesan area.
There are 165 Catholic primary and post-primary schools between the diocese of Dromore and Armagh.
He told the BBC: "We've got time, it's time now to suspend these transfer tests. These are 10-year-old children.
"I'm calling on the whole educational community to suspend the use of academic selection this year.
"I'm asking our grammar schools to publish alternative admissions criteria that don't rely upon use of transfer tests that would be carried out in such uncertain and challenging circumstances."
The Bishop of Derry, Donal McKeown, has sent a similar letter to grammar schools in the city in his role as a trustee of Catholic schools in his diocese.
The transfer tests are to be held two weeks later than usual in 2020 due to the coronavirus pandemic.
That has been criticised by some MLAs and some primary principals.
In his letter, which has been obtained by BBC News NI, Archbishop Martin said he was "aware of the anxiety that is already building over the prospect of Primary Seven transfer tests in the context of the Covid-19 pandemic".
"In recent weeks, parents and teachers have spoken to me of their disquiet and about the uncertainty that is already upsetting some children," he added.
He said that parents and teachers were doing their best in the circumstances, but some pupils were struggling without access to technology or the help of their teachers.
Archbishop Martin also said schools would face huge challenges in the event of pupils coming back on a phased basis in September.
"Pupils may require additional personal and emotional support with coming back to school after such a long and stressful break from the classroom," he said.
"With all this in mind, and in the best interests of our children and schools, I appeal to the whole educational community in the Archdiocese of Armagh to support a suspension of the use of academic selection for entrance to post-primary schools in September 2021.
"Our Catholic schools rightly take pride in a caring ethos which places Christ, and the well-being of our children, at the centre of everything that we do."
"In making this appeal for a suspension of the use of academic selection in the coming year, I have no desire to simply re-open the arguments for, and against, the transfer tests.
"I make this call from my heart on behalf of the current Primary Six children, their families and all our school communities," he concluded.
In his separate letter - also obtained by BBC News NI - Bishop Donal McKeown said that socially disadvantaged pupils would be "seriously disadvantaged" in the transfer tests as they would have been out of school for almost six months.
"Therefore, in the interest of fairness to pupils and in the interest of maintaining a mix of pupils in the school community, I would ask your board of governors to consider how you could respond to this," he wrote.
"This may well be a situation where criteria other than transfer test grades should be included for at least a substantial proportion of the 2021 entry cohort."
Bishop McKeown also said that he would be happy to discuss a way ahead with the grammar schools in the diocese.