In the poll, commissioned by Heathrow Airport, 49% supported expansion and 32% opposed it.
Heathrow say the poll is the biggest undertaken locally since the Airports Commission was formed. At least 1,000 residents in each constituency were polled, 9,670 in total.
As well as general support, 25% of local residents are more likely to vote for a Parliamentary candidate if they support expansion, while only 17% are likely to vote for them if they are against the plans.
59% of those surveyed said they feel positive towards Heathrow, while 34% feel neutral and just 6% feel negative towards the airport.
The airport say the research is significant because it covers a statistically representative sample of those who stand to be most affected by Heathrow expansion and who have traditionally been perceived as opposed to growth.
Only one constituency – Richmond – had more residents who opposed expansion than supported it, consistent with earlier surveys.
John Holland-Kaye, Chief Executive at Heathrow said: “The whole country can get behind this plan. Expansion at Heathrow will connect all of Britain to global growth, and keep Britain at the heart of the global economy. We have listened and improved our proposal so that it benefits local communities. Expansion at Heathrow can deliver 50,000 new local jobs, tackle youth unemployment and add 5,000 new apprenticeships.
“We are offering a fair deal by significantly increasing our funding for compulsory purchase and noise insulation for local residents. Noise at the airport will reduce compared to today and we can stay within the UK’s environmental limits.”