The season was suspended in March in response to the novel coronavirus outbreak but clubs were permitted to begin close contact training this week.
"Players or club staff who have tested positive and their close contacts will now isolate and be assessed in line with the PHE-agreed guidelines," the Premiership said.
"No specific details as to clubs or individuals will be provided."
The other four positive tests were from non-playing staff.
The Premiership, which has nine rounds remaining, is hoping to resume on the weekend of August 14.