The club cancelled training on Monday as a precautionary measure and also scrapped a planned media conference ahead of Saturday’s Rainbow Cup match against Ulster.
Edinburgh hope that the game can still go ahead at BT Murrayfield as scheduled but it will depend on further results. The entire squad was tested on Monday morning.
The club have not named the player who tested positive due to medical confidentiality but have confirmed he is isolating along with “a small group of potential contacts”.
A statement from the club said: “Edinburgh Rugby can confirm a squad player has tested positive for Covid-19.
“The player has started self-isolation as per Scottish Government guidelines.
“Subsequent internal contact tracing identified a small group of potential contacts and they are also isolating. To respect medical confidentiality Edinburgh Rugby will not name the individuals affected.
“Today’s (Monday) training at the club has been cancelled as a precaution and the entire squad and playing staff were tested early this morning (Monday). Further updates on the squad will be issued as appropriate.
“The club continues to operate an extensive health and safety coronavirus mitigation plan and follow all respective health guidance aligned through Scottish Rugby's Threat Management Group.
“The club’s Guinness Pro14 match against Ulster at BT Murrayfield this Saturday (5 June) is still scheduled to go ahead as planned.”
The Rainbow Cup has already lost one match to coronavirus.
Ulster’s game against Ospreys scheduled for last Saturday had to be cancelled due to positive Covid-19 cases in the Irish club’s squad.
With no available weekends remaining for the game to be rescheduled, four match points were awarded to Ospreys.
Pro14 Rugby, the organisers of the Rainbow Cup, said this was as per the protocol decided prior to the start of the 2020/21 season whereby a game that could not be rescheduled would result in the fixture being deemed a 0-0 draw, but four match points would be awarded to the team who had not been the cause of the postponement.
"This protocol ensures that no artificial scoring points were added to the tries/points for and against columns,” said Pro14 Rugby in a statement.