The outbreak took place among employees at vegetable producer AS Green and Co, which is located in the village of Mathon near Malvern.
A joint statement from Public Health England (PHE) Midlands and Herefordshire Council said that the workers were being told to remain on site for the duration of their time in quarantine.
Officials are treating the group as "one extended bubble", since all the members of the workforce live in mobile homes on the farm during the harvesting season.
The council added that it was arranging deliveries of food and other supplies to the residents during their self-isolation.
Katie Spence, PHE Midlands health protection director, said that infection control measures had been in place at the farm.
Ms Spence said in a statement: "Despite these measures, a small number of workers became symptomatic earlier this week and they and a few close contacts among the workforce were tested initially and found to be positive."
She added that the whole workforce was then tested as a precaution and that almost all of the results have now come back.
In response to the outbreak, a spokesperson for AS Green and Co said it was prioritising its "hard-working" staff.
A statement on the farm's website confirmed that it was closed to visitors and that the workers would not leave the farm until they had completed their period of self-isolation.
"Our work force and local community are our priority at this difficult time and we continue to follow the guidance of the relevant bodies to ensure that the spread of the virus is controlled and our workforce is supported," the statement said.
"Public Health England advises that it is very unlikely Covid-19 can be transmitted through food or food packaging, so shoppers can remain confident buying British fruit and veg," it added.