The new runway would allow 260,000 more flights each year when completed in 2029, the BBC reported Wednesday.
London Mayor Boris Johnson has opposed the plan, saying the expansion would be "disastrous" for residents of the capital city.
The proposed runway would be built to the north, northwest or southwest of the current airport. The plan was developed at the behest of the Davies Commission, which is examining the future needs of British airports.
In the airport's submission to the committee, Heathrow chief executive Colin Matthews said the added runway would increase the airport's capacity to 740,000 flights a year from the current limit of 480,000.
He said a fourth runway might be required, but not until at least 2040.
The London mayor called the proposed new runway "politically, environmentally and socially unacceptable."
"There will be more pigs flying than aircraft if we are to believe the claim that three runways at Heathrow will make less noise than two," he said.
Johnson published his own proposals for alternative hubs Monday: an artificial island on the Thames Estuary, a major expansion of Stansted Airport and construction of an airport on the Isle of Grain in north Kent.
The Davies Commission is expected to issue its final recommendations about airport expansions during the summer of 2015.