The Department of Health said it had ordered an investigation.
It said in a statement: “NHS Test and Trace is processing tests at an unprecedented scale – 225,000 a day on average over the last week – and the vast majority of people going through the system do not experience any problems.
“Everyone who tests positive is contacted directly and informed of their results either by text or email. We are not aware of this issue and will investigate.”
A Department of Health source added: “Individuals are required to provide a phone number or email address only to order a Covid-19 test.
“In a minority of cases, we do not have the correct information for both methods for contact tracers to use. For example, an individual may order a test via email but we may not have the correct phone number, or vice versa.”
An ICO spokesman said: “People’s medical data is highly sensitive information, not only do people expect it to be handled carefully and securely, organisations also have a responsibility under the law to ensure that it is handled appropriately.
“When anyone is concerned about how their personal information has been handled, they should first contact the organisation involved, if they are still not satisfied they can bring it to the ICO.
“Organisations also have a legal duty to report certain breaches to the ICO.”
An ICO source added: “If a data incident occurs, we expect organisations to consider whether it is appropriate to contact the people affected, and to consider whether there are steps that can be taken to protect them from any potential adverse effects.
“If someone is concerned about how their personal data has been handled and makes a complaint to us, after having first contacted the organisation, we will assess the information provided.”