Airport bosses believe adding 25,000 more flights to the existing annual limit of 480,000 would lead to a "Brexit boost" worth £1.5bn to the UK economy between 2021 and the opening of a third runway in 2025.
This increased capacity would be ring-fenced to create up to 21 new daily domestic services and 13 long haul routes, according to Heathrow.
The measure could mean new routes to UK destinations such as Dundee, Newquay and Liverpool, and to growing international markets including the Japanese port city of Osaka, Ecuadorian capital Quito and central Chinese city Wuhan.
John Stewart, chairman of anti-Heathrow group Hacan, said: "The cap on flights has always been regarded as sacrosanct by residents but if it is to be lifted in advance of a third runway, it is essential that measures such as a tougher night flight regime are in place.
"It's very clear that this package has been rushed out in response to the Government's concern that a third runway won't be ready in time to deliver improved connectivity post-Brexit.
"Heathrow has been under real pressure to show that it can do something since both the rival schemes - a second runway at Gatwick and the extended runway at Heathrow proposed by Heathrow Hub - can be built more quickly and cheaply."
Ministers are currently considering which project to support and a decision is expected in the coming weeks.
Heathrow's latest proposals, to be launched in full at the upcoming Conservative Party Conference in Birmingham, also suggest extending the airport's £10 discount for every departing passenger flying to a UK destination for 20 years.
The west London hub claims its plans will enable the whole country to capitalise on new market opportunities in the early months of the UK leaving the European Union, supporting small and large exporters, boosting competition and reducing prices for consumers.
The airport published research by consultancy Frontier Economics which showed that - assuming the UK leaves the EU in 2020 - it will create around £55bn more growth than an expanded Gatwick in the 15 years after Brexit.
Heathrow chief executive John Holland-Kaye said: "This Brexit boost plan is our commitment to making Britain stronger and fairer for everyone - faster.
"Our proposals for an additional 25,000 flights a year from 2021 would help businesses and families from Newquay to Dundee benefit earlier from Heathrow expansion, while protecting our commitment to meet and exceed the Airports Commission's environmental conditions.
"Heathrow's third runway is the only option that can help every nation and region of Britain realise the opportunities of Brexit. The Prime Minister and the government can now make the right choice, and back Heathrow expansion."
Liverpool John Lennon Airport boss Andrew Cornish said an expanded Heathrow would create the opportunity for flights to operate between the airports, linking Liverpool with "every continent on the globe".
Mike Cherry, Federation of Small Businesses national chairman, believed Heathrow's proposals for boosting capacity while a third runway is built would be "warmly welcomed" by small businesses across the UK.
Gatwick has previously claimed it is the only expansion project that can be delivered.
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