One traveller said the Tui flight was full of "covidiots" and "inept crew who couldn't care less". Another claimed there "wasn't much" policing of rules.
Sixteen passengers on flight TOM6215 from the Greek island to Cardiff have now tested positive.
Tui said safety was a priority and it was concerned by the claims.
"Our crew are trained to the highest standards," the airline said. "A full investigation is now under way as these concerns weren't reported during the flight or before today."
The 193 passengers and crew who were on board the flight on 25 August are being asked to self-isolate for a fortnight.
Health officials said seven people from three different parties are believed to have been potentially infectious on board the aircraft.
Public Health Wales' Dr Giri Shankar told BBC Radio Wales people returning from Zante in particular are "picking up the infection and are coming to our attention".
Greece and its islands remain on the UK government's list of quarantine-exempt countries, meaning travellers do not need to self-isolate on their return.
But the Foreign and Commonwealth Office (FCO) updated its travel advice for Zante on 29 August to reflect local coronavirus restrictions put in place on the island.