Sandwell Council said one third of the workforce at CBS Packaging in West Bromwich have been affected by the disease but has not yet confirmed whether than means one third of staff have tested positive.
CBS employs employs around 200 people across its sites in West Bromwich and Essex, with the majority thought to be based in the Black Country.
The firm, based on Brandon Way, is to start testing all of its workforce from Monday.
It comes after a spike of Covid-19 cases in neighbouring town Smethwick was reported on Thursday.
Dr Lisa McNally, Sandwell's director of Public Health said: "The company has told people to isolate at home if they have symptoms.
"They are taking the advice of public health partners on safety and infection control measures.
"We’re very pleased that the company encouraged their staff to take the tests which has enabled them to identify these cases.
"I would encourage other companies to come to us if they also have any concerns about COVID-19."
There are concerns Covid-19 could be spread in Smethwick and Walsall as a result, said Dr McNally.
However she said both events - the cases at CBS and the spike seen in Smethwick - are likely not directly linked, adding: "We know the virus does not respect geographical boundaries and people do travel all over the Black Country and Birmingham for work."
The firm's managing director Jitha Singh said: "We very much appreciate the help and support from the public health teams to help us deal with this situation and try to get it under control.
"Obviously we have tried to do everything we are meant to – but this is new to all of us and we are all learning.
"As a businessman I would encourage other businesses to take up all the help available. Pick up the phone and ask for help – don’t put it off."
CBS is not the first Black Country firm to be badly hit by coronavirus - last month around a dozen staff tested positive at Tulip Foods in Tipton.
According to the latest Government figures, 1,682 people have tested positive for coronavirus in Sandwell since the pandemic began while 5,881 people have tested positive across the four Black Country boroughs.