The Irish club had to shut down their high performance centre in Limerick on Tuesday following the positive test.
Six other players identified as potential close contacts are now self-isolating.
The unnamed player who tested positive was not in the travelling party for the match in Wales against Scarlets last weekend.
A further round of testing took place on Wednesday and Munster released the following statement: “Munster staff and players underwent a round of PCR testing at the HPC this morning and will await results ahead of a scheduled return to training on Thursday.”
The match with Edinburgh is due to take place at Thomond Park on Saturday evening.
Munster also announced that two of their players had sustained serious injuries.
Hooker Niall Scannell is out for two months and will have to undergo surgery after damaging his neck at training last week.
Neil Cronin sustained an anterior cruciate ligament injury at training on Monday and will meet with the specialist on Friday to discuss surgery.
The Irish province’s casualty list also includes Chris Cloete (neck) and Keith Earls who is rehabbing a back injury.
There is better news on Conor Murray and Shane Daly with both players expected to train this week after thigh and groin issues, respectively.
Munster won their season opener against Scarlets 30-27 in Llanelli while Edinburgh lost 25-10 at home to Ospreys.