It comes as prisons in the North of Ireland prepare to suspend all visits from 6 pm on Friday, October 16.
The move, which also happened during lockdown in March, will last four weeks, in line with wider restrictions being brought in by the executive.
There are 1,500 prisoners spread across three sites in the North of Ireland – Maghaberry, Magilligan in Limavady an Hydebank Wood in Belfast.
In a statement, the Prison Service said a Maghaberry prisoner reported a high temperature on Tuesday and was “immediately tested and placed in isolation” pending the result.
It came back positive and track and trace procedures were implemented.
Other prisoners accommodated on the same jail landing are “being supported by staff and monitored by medical staff from the South Eastern Trust”.
The statement added one member of staff had also been advised to self-isolate.
The temporary suspension of visits is unconnected and follows discussions with Justice Minister Naomi Long.