NHS Lanarkshire has contacted all parents of children as well as members of staff at Firtrees Nursery in Motherwell where the person works, to give them advice and support.
The health board said the individual concerned was well and had been responding to treatment.
Some children and staff who may have come into contact with the person were being offered screening as a precaution in line with national guidance.
Dr Josephine Pravinkumar, NHS Lanarkshire consultant in Public Health Medicine, said: “Tuberculosis (TB) is a relatively uncommon infection in Scotland and is not easily transmitted from person to person.
“It can be successfully treated with antibiotics.
“The risk of transmission of this infection to people who have been in contact with a case is low and the risk of people developing active TB disease is even lower.
“You can only catch TB from prolonged contact with someone who is coughing up TB germs although prolonged contact does not necessarily mean you will get TB.
“All parents and staff have been provided with information and support and, as a precaution, screening is being offered to those we’ve identified as close contacts.”
It is thought the member of staff is recovering at home after testing positive in the last week.
A dedicated helpline has also been set up for parents and staff on 0800 0 282 836.
TB is a disease which usually affects the lungs but can affect other parts of the body.
Symptoms can include a cough, sometimes with blood-stained spit, unexplained weight loss, fever or sweating, especially at night, and shortness of breath.
It is spread by the person with TB coughing the bacteria from their lungs into the air and another person breathing in air containing bacteria.