The Yorkshire Post understands that Transport Secretary Chris Grayling is preparing to table the national policy statement on a third runway within the next two weeks, after which a vote must be held within 21 days.
Long-term opponents of expansion such as Tory Foreign Secretary Boris Johnson and Labour Shadow Chancellor John McDonnell threaten to cause a headache for their parties by voting against expansion.
It is understood that neither party has decided exactly how MPs will be instructed to vote.
The Prime Minister may seek to come up with some sort of formulation to allow Tories with longstanding and local opposition to the project to oppose it, in order to avoid Mr Johnson breaching Cabinet collective responsibility.
The Foreign Secretary, whose Uxbridge and South Ruislip constituency is near the west London airport, once said he would lie down “in front of those bulldozers” to stop a third runway being built.
High-profile west London MPs such as Justine Greening and Zac Goldsmith are also almost certain to vote against a third runway.
A decision on Heathrow expansion was repeatedly delayed under David Cameron’s administration despite the Airports Commission in 2015 calling for a new runway at the airport, underlining the political sensitivity of the issue.
When Mrs May announced in October 2016 that the Government would back expansion, Mr Goldsmith dramatically quit and forced a by-election in Richmond Park.
Standing as an independent, Mr Goldsmith lost to the Liberal Democrats before returning to the Commons as a Tory in last summer’s general election.
Labour backs airport expansion in the south-east but sources have said the party will study the policy statement before deciding how to vote, arguing it will need to satisfy four tests on noise, air quality, climate change and economic growth.
Heathrow Airport bosses are understood to be confident the proposals will meet the tests.
But Mr McDonnell is believed to think it is “very unlikely” the plans will pass the four tests and could try and persuade Mr Corbyn to whip MPs against a third runway.
Heathrow Airport has argued that a third runway could deliver up to £12bn in economic benefits and 11,200 jobs in Yorkshire, with 11 sites in the region in the running to be offsite construction centres for a new runway.
Heathrow officials have visited Leeds Bradford Airport, Doncaster Sheffield Airport and other locations to scope out potential “logistics hubs”, where components of the new runway will be pre-assembled before being transported to west London.
Emma Gilthorpe, Heathrow executive director of expansion said: “Heathrow expansion will be beneficial to the whole of the UK delivering increased trade, tourism and employment.
“It will be one of the UK’s largest, privately funded infrastructure projects that communities nationwide, will prosper from.
“Heathrow is the UK’s only hub airport, helping to connect Yorkshire to destinations across the globe; connectivity which is only set to grow from an expanded airport.”
“We are looking forward to the Government tabling a vote on expansion soon.”