The body has opened enforcement cases into both airlines and written to them detailing its concerns.
It is now seeking to resolve these concerns with the companies.
The CMA said it would not provide any additional public information about the cases and it should not be assumed at this time that either airline has broken the law.
“Ultimately, only a court can decide whether a breach has occurred,” added a statement.
The investigation will consider situations where airlines continued to operate flights despite people being unable lawfully to travel for non-essential purposes in the UK or abroad, for example during the second lockdown in England in November.
Rory Boland, Which? Travel editor, said: “Millions of customers were treated appallingly last year when airlines continued to operate flights during lockdown restrictions but refused to refund their passengers for flights they could not legally take – so it is right that the CMA is investigating two of the largest airlines for potentially breaching consumer law.
“Customers should not have been penalised for simply following the law and trying to protect public health, so if the regulator finds that Ryanair and BA have breached consumer law, we would expect to see all affected customers given the option of a refund.