The Competition and Markets Authority said it had opened a probe into one firm, Expert Medicals, following complaints Covid test results had not been provided on time or at all.
It warned a further 19 firms about advertising misleading prices.
It comes after a review that found customers often faced poor service, hidden fees and refund issues.
UK travellers must take PCR tests for Covid when they return from foreign destinations, choosing from a list of 400 or more authorised government providers.
However, holidaymakers have complained of high prices and missing tests or results from many of the firms, prompting Health Secretary Sajid Javid to call for a review into "excessive" pricing and "exploitative practices".
On Friday, the CMA said it had begun investigating Expert Medicals, in Bradford, following a large number of complaints to Citizens Advice and the regulator directly.
The firm, which was one of the UK's largest PCR travel test providers this summer, allegedly failed to supply Covid tests or respond to customer complaints and has refused to issue refunds when due.
"It now no longer appears on the list of testing providers who self-declare that they meet the government's minimum standards for Covid-19 testing," the competition watchdog said.
The CMA said 19 other firms may be falsely advertising tests at very low prices when they are either not available at that cost or include hidden conditions, such as where the tests must be collected from.
Andrea Coscelli, chief executive of the CMA, said: "Millions of people have to buy PCR tests in order to travel. If we find that firms have been misleading customers and failing to provide the service that people have paid for, we stand ready to take action.
"Our actions today are the next steps in our work to ensure the PCR testing market works for consumers and we are preparing to take further action in this sector."
Tests cost about £75 per person on average in the UK, but there are reports of some being offered for as little as £20, and others more than £500.
There have also been reports that some government authorised providers have not got full accreditation.
Earlier this week the government said it was warning 82 firms that they could be removed from its list of authorised providers if they advertise misleading prices.
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