The school said it had been in contact with Wiltshire Council’s Public Health Team and has established the close contacts of the individuals.
Children who have been in close contact with both members of staff have been contacted directly and have been told to stay at home for 14 days.
Steve Heal, headteacher of Malmesbury CE Primary School, said: “We have become aware that two members of our staff who live in the same household have tested positive for Covid-19.
“This follows another member of the household having tested positive earlier, so this is almost certainly household transmission, not in-school transmission.
“The staff members developed symptoms on Saturday.
“One of them was not in school from Wednesday lunchtime, so was not here while infectious and contacts will not need to isolate.
“However, the other was in school on Thursday from lunchtime (but not on Friday) so contacts will need to isolate for 14 days.
“We have conducted a thorough risk assessment with Wiltshire Council Public Health Team to identify close contacts and any additional actions required.
“The parents/carers of pupils identified as close contacts have already been notified in a separate communication.
“Unless you have received a separate email, your child does not need to isolate.
“Whilst I fully understand this will cause anxiety amongst the Malmesbury community, I can assure you all necessary steps are being taken to minimise risk to others, to cause the least disruption possible and to support all those affected.”
The school remains open to the majority of students.
If a pupil or another member of the household develops symptoms of Covid-19, they should arrange a test via bit.ly/34eqENX or by calling 119 and remain at home with the rest of the household while waiting for the result.