More coronavirus cases have also been confirmed at a school, nursery and a pub in the area.
It comes after the local authority warned it could be the first area in Wales heading for a local lockdown.
A walk-in testing facility was set up in response on Saturday, but by Monday morning Aneurin Bevan University Health Board warned the facility was already busy.
On Monday it was confirmed two staff members at the Crossways Tesco had tested positive for the virus.
A second school in the area has also been affected by a coronavirus case with the class being required to self-isolate after a pupil tested positive.
Two cases of Covid-19 at the Crossways Tesco were confirmed on Monday. It is understood the individuals caught the virus in the community, not while at work.
“We have introduced extensive measures across all of our stores to help keep everyone safe, including protective screens at every checkout, social distancing signage and regular deep cleaning.”
It comes as the health board urged people to attend the walk-in facility in the afternoon after experiencing “high levels of demand” on Monday morning.
A member of staff at St Gwladys Primary School in Bargoed tested positive for the virus, with 21 pupils from one class being required to self-isolate for 14 days.
A class at a second school, Bedwas Infants School, have also been advised to self-isolate after one pupil tested positive for the virus.
In a note sent to parents on Sunday, the school said: “A pupil has tested positive for coronavirus and we therefore need to follow strict public health protocols to ensure the health and wellbeing of the whole school community.
As a result, all children in Porffor B (Those in on Friday September 4) and the staff who worked with Porffor B will need to isolate at home for 14 days.
A spokesman for Caerphilly County Borough Council said: “We are aware that a pupil at Bedwas Infants School has tested positive for Covid-19 and that the school is taking all appropriate steps to ensure the health and wellbeing of pupils and staff.
“The school is following Public Health Wales guidance and those pupils and staff that are required to self-isolate have been advised to do so.
“The school will remain open and the headteacher is in regular contact with parents and is keen to reassure everyone that robust risk assessments and procedures are in place to deal with these types of incidents.”
A pub in Bedwas, the Fisherman’s Rest, also reported one member of staff testing positive for the virus.
Jonny Harris, the pub’s landlord, posted on Facebook that a member of the bar staff felt ill and took a drive through test which resulted in positive.
He said: “I have a pod system in place and all of those in that bar staff pod have been advised, by me, to remain away from the pub until a test a result is provided.”
He said track and trace details were available for the NHS and confirmed that the pub would remain closed until at least Wednesday.
He added: This will allow for many to go to the temp testing station at Caerphilly leisure centre which is due to end Tuesday Evening and isolate pending their results
“Although there are several control measures in place at the pub, which will be enhanced on opening, there is a significant amount of personal responsibility for everyone to play their part when they visit.”
A private nursery in Bedwas also posted a statement to their Facebook page reporting they had been informed that a nursery child in their preschool room received a positive coronavirus test result.
The statement, posted on Sunday, said: “We have been informed that a nursery child in our Preschool Room was yesterday tested for Covid and has received positive test result today.
“The source of the spread of the virus has been traced back to a family member who the child had contact with on Wednesday September 2. The child last attended nursery on Thursday September 3.
“We are in the process of contacting the parents and carers of children who attended in the Preschool Room on Thursday September 3, along with any staff members who provided care to the children on that day.”
A deep clean of the nursery was undertaken on Sunday and the nursery reopened on Monday.
World news – GB – Two Tesco workers test positive for coronavirus in Caerphilly