The investigators are focused on a call centre in the region that carries out contact tracing work for Public Health England.
According to the BBC, one employee said he thought at least seven of his colleagues had tested positive for the virus.
NHS Lanarkshire said it was aware of a number of "potentially linked cases".
NHS Lanarkshire consultant in public health medicine, Dr David Cromie, said: "We became aware this morning [19 July] of a number of potentially linked cases of coronavirus (Covid-19) in Lanarkshire.
"We instigated some immediate measures to reduce risk and are currently investigating the situation."
The news comes after the Scottish government confirmed another rise in the daily coronavirus figures this afternoon.
The number of new coronavirus cases in Scotland rose to 23 - the highest daily figure since 26 were reported on June 21
Cases have steadily risen each day since Tuesday, when three were reported.
Reacting to the latest update, First Minister Nicola Sturgeon tweeted: “This is still a low number of cases - and fluctuation is to be expected - but they will all be carefully examined, so that any patterns are identified.
“Test & Protect is also working to break the chains of transmission,” she added.
There have now been a total of 18,455 positive cases of coronavirus in Scotland since the start of the outbreak.
302 of those were being treated in hospital as of last night - three in intensive care, along with another person who is suspected to have the virus.
The total number of people who have died from Covid-19 stands at 2,491.
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