According to reports, British Airways has stopped selling short-haul tickets from the airport.
Tens of thousands of flights have been canceled this summer as the industry struggles to cope with demand.
According to the Times, British Airways suspended tickets for domestic and European services until Monday to comply with the cap on passenger numbers at the airport.
A million passengers will be affected by the cancellation of 10,300 flights by the company.
Many passengers flying to and from the UK's busiest airport have been disrupted recently with long security lines and baggage system failures.
The airport was accused of showing disregard for consumers by trying to force it to deny seats to tens of thousands of travellers through the cap.
If any airline wanted to put profit ahead of a safe and reliable passenger journey, it would be disappointing.
Virgin Atlantic said that the airport was to blame for the chaos because of its failures.
On 21 July, airlines were accused of treating passengers unfairly.
The Competition and Markets Authority and the Civil Aviation Authority are concerned that consumers could experience significant harm if airlines don't meet their obligations.
Some airlines may not be doing everything they can to avoid engaging in harmful practices, according to the letter.
Selling more tickets for flights than they can reasonably expect to supply, not always "fully satisfying obligations" to offer flights on alternative airlines to passengers affected by cancellation, and failing to give consumers "sufficiently clear and upfront information about their rights" are examples of these.