The Airports Commission will recommend Heathrow as the preferred site for London's new runway, sources have told the BBC.
But while backing Heathrow, it may also leave the door open for alternative expansion at Gatwick.
The commission, led by Sir Howard Davies, is due to make its recommendation at 07:00 on Wednesday.
Downing Street said it would not be be making "a snap judgement" in response to report.
Number 10, which was given a copy of the report on Tuesday evening, said it wanted to "properly read and properly digest" the report. A formal government response is not expected until the Autumn.
With no clear decision, the report is likely to open up splits within the Conservative party over which airport to back.
As the commission is an independent body its report does not bind the government.
A new runway at Heathrow would affect the constituencies of at least five Conservative cabinet ministers, including Theresa May, Phillip Hammond, Greg Hands, Justine Greening and Theresa Villiers.
London Mayor Boris Johnson and Richmond Park MP Zac Goldsmith, who is bidding to replace him, are also opposed to the Heathrow site.
Residents near the airport have already launched legal action against the way the commission has considered the pollution and noise impact of a new runway.
By Kamal Ahmed, BBC business editor
After decades of delay and endless reports, the government's latest attempt at sorting out Britain's airport capacity problem is set to back the building of the first new full-length runway in the South East of England since the 1940s.
Sir Howard Davies's Airports Commission was asked by David Cameron to come up with recommendations on how to satisfy the ever increasing demand for air travel. Both Heathrow and Gatwick are operating at or close to capacity.
Sources in the aviation industry and across Whitehall believe Sir Howard will back a third runway at Heathrow, saying it will bring significant economic benefits to the UK.
He will not, though, rule out a second runway at Gatwick as a credible option - meaning that the government will be left to make the final decision.
Business leaders have thrown their support behind Heathrow, saying that Britain needs a flourishing hub airport to connect the UK to emerging markets around the world.
But Mr Cameron may find it particularly difficult to support the move after ruling it out in 2009, saying the pledge was made with "no ifs, no buts".
Whatever the final decision, building a new runway, if it ever happens, is likely to take at least a decade.
A third runway at Heathrow was proposed by the last Labour government but the plan was scrapped when the Conservatives and Lib Dems formed their coalition government in 2010.
The commission, set up in 2012 under Sir Howard Davies, has considered three options for additional runway capacity in the South East of England which it says is needed by 2030.
- A third runway at Heathrow
- Lengthening an existing runway at Heathrow
- A second runway at Gatwick
However, the commission's report was put off until after May's general election.
The Conservative Party's 2015 manifesto promised to "respond" to the final report but government sources told the BBC earlier this month there would be no immediate response to the final recommendation. The Financial Times has reported that there will be no decision until Christmas.
Spokesmen for Gatwick and Heathrow declined to comment.
Green campaigners maintain a new runway will harm the environment.
Friends of the Earth's head of campaigns Andrew Pendleton said: "It's simply not credible for the government to build a new runway in the South East and still claim to be serious about tackling climate change.
"Airport expansion will also have huge impacts on the local community, noise levels and air quality. We can't preach to the world about stopping catastrophic climate change on the one hand and send aviation emissions soaring on the other."