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The unnamed woman, who is 22, said she was contacted by the NHS Test and Trace worker in June.
They were checking to see if she was self-isolating, as she had returned to the UK from Italy.
However, she says the call handler ended up reaching out to her on Instagram just one hour later, allegedly saying: "Gonna sound weird, but spoke to you thru NHS..."
The man told her she seemed 'cool' and 'pretty', asking her where she was from.
Speaking to The Daily Mail, the woman - who opted not to share her name - said: "This man acted in a predatory manner and has gone out of his way to track me down and send me creepy messages though my private Instagram page."
The language student said the incident left her feeling shaken, as the call handler knew her home address.
She added: "Someone like that should not be working for NHS Test and Trace, where they can access such sensitive personal data... there was the possibility that his online stalking could progress to physical stalking."
The Department of Health and Social Care has since said it will investigate what happened, saying it suspends staff when necessary.
A spokesperson told LADbible: "NHS Test and Trace is committed to protecting the personal data and safety of people they contact.
"All staff must adhere to the highest professional standards and if staff fall below these standards we will not hesitate to take immediate action to suspend individuals, where appropriate."
The incident took place before travel restrictions changed today (19 July), as part of the government's roadmap out of lockdown.
From today, people returning to the UK from amber list countries such as Italy will no longer need to quarantine, so long as they are fully vaccinated.
However, this change in rules does not currently apply to travellers returning from France.
Travellers will still be required to take a pre-departure test three days before they leave for England, along with another test on or before the second day after their return.
Find out more about all the changes coming into place today, including rules on masks and social distancing, here.
Further down the line, from Monday 16 August, those who have had both doses of their vaccine will also be exempt from self-isolating if they are found to have come into contact by someone who later tested positive for Covid by NHS Test and Trace - unless they test positive for coronavirus, of course.
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