Transport Minister Chris Grayling will make a statement to parliament about the expansion at 1130 GMT on Tuesday.
"There will be challenge and opposition whatever option we take," he told BBC television on Sunday, of the first full-length runway to be built in the London area for 70 years.
"We have to, in my view, take a decision that is in the interests of our nation, what delivers the best connectivity, the right approach for the future at a time when we want to grow international trade links, open up new opportunities."
The decision will be subject to public consultation before it is put to a vote in parliament in late 2017 or early 2018. Surveys show a majority of lawmakers back Heathrow expansion.
Five local councils around Heathrow including Maidenhead, the area May represents, have hired a law firm to fight Heathrow expansion. Other opposition groups have also said they are likely to consider legal action against a new runway there.
Costing either $17 billion for the runway extension or $22 billion for the additional third runway, Heathrow would be the more expensive project and face legal challenges over its environmental impact on densely populated west London.
But its established trade links with emerging markets strengthen its case in the wake of the Brexit vote. It also has the backing of the major airlines.
Gatwick, the country's no. 2 airport that mostly connects to Europe, argues it can build a runway more quickly and, at $9 billion, more cheaply, and that its rural position means it would disturb fewer people.
Writing in the Telegraph newspaper, Howard Davies, head of the independent inquiry which backed Heathrow, said Brexit had made the case for Heathrow "overwhelming" given it handles more air freight and a greater number of long-haul inbound tourism.