Writing as Archbishop of Armagh, the Derry-born Archbishop Eamon Martin 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.
Archbishop Martin said that testing in the current climate was “cruel”.
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 two dioceses.
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 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 that 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.”
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