BBC Scotland understands that the outbreak involves a call center that searches for coronavirus contacts for Public Health England.
An employee said he believed at least seven of his colleagues were positive.
The NHS Lanarkshire has stated that it is aware of a number of “potentially related cases”.
Dr David Cromie, NHS Lanarkshire consultant in public health medicine, said: “We realized this morning [19 July] of a number of potentially related cases of coronavirus (Covid-19) in Lanarkshire.
“We have initiated some immediate measures to reduce risk and are currently studying the situation.”
Previously, the Scottish government claimed that 23 new Covid-19 cases had been detected in Scotland in the past 24 hours, although only three of them were in the Lanarkshire health council area.
This is followed by an increase of 21 new cases on Saturday, the highest daily figure in almost a month.
Prime Minister Nicola Sturgeon said that the number of cases is still low and that the fluctuations are “foreseeable”.
The percentage of positive tests remains well below 1%.
Earlier, health officials said they did not believe they were dealing with a related cluster as new weekend cases, which appeared in official statistics, were spread across a large area of Scotland.
Public Health England was contacted for comment.