British Airways and Ryanair may have broken consumer law in not offering refunds to customers who couldn't legally take flights due to Covid lockdowns, a watchdog has said.
The Competition and Markets Authority (CMA) said the airlines' stance could have left customers out of pocket.
The airlines offered other options, but both refused refunds, it said.
However, BA said it had acted lawfully and Ryanair said it had refunded customers in justified cases.
During the three UK lockdowns people were told to avoid all non-essential travel.
The investigation is looking at flights that weren't cancelled, but that people legally could not go on due to lockdown restrictions.
The CMA said that while BA had offered vouchers and rebooking for flights that weren't cancelled, and Ryanair had offered rebooking, both had declined refunds.
Andrea Coscelli, chief executive of the CMA, said: "While we understand that airlines have had a tough time during the pandemic, people should not be left unfairly out of pocket for following the law.
"Customers booked these flights in good faith and were legally unable to take them due to circumstances entirely outside of their control. We believe these people should have been offered their money back."
The watchdog is concerned that during lockdowns in the UK, when most people were legally unable to fly, the airlines failed to offer refunds
It said it wanted to resolve those concerns, which might include the airlines offering affected customers refunds "or other redress".
The CMA has sent a letter to the companies outlining its concerns.
Usually the watchdog tries to work with firms to address any issues.
However, if the firms push back, it can seek a court order to compel them to refund customers and make changes to their practices.
The CMA cannot fine the firms if it is found they breached consumer law - which is different to its powers in competition cases, where it can levy fines.
But BA said it offered "highly flexible booking policies at the same time as operating a vastly reduced schedule due to government-imposed travel restrictions, and we have acted lawfully at all times".
A spokesperson said: "During this unprecedented crisis we have issued well over three million refunds and helped millions of our customers change their travel dates or destinations and we're grateful to them for their ongoing support."
"It is incredible that the government is seeking to punish further an industry that is on its knees, after prohibiting airlines from meaningful flying for well over a year now.
"Any action taken against our industry will only serve to destabilise it, with potential consequences for jobs, business, connectivity and the UK economy."
Ryanair said it had looked at refunds on a case-by-case basis and had paid money back "in justified cases".
"Since June 2020, all our customers have also had the ability to rebook their flights without paying a change fee and millions of our UK customers have availed of this option," it said in a statement.
The CMA has been investigating various parts of the travel industry since July last year.
It opened its investigations after getting complaints about refunds, cancellations and price rises following the first lockdown in March.
The watchdog has received 148,000 complaints overall so far, with many thousands of those being about holiday refunds.
Then in May it warned travel companies
that they needed to be ready to refund customers, and said that five big package holiday firms including Tui, Lastminute.com and Virgin Holidays had refunded a total of £200m.