Mrs May's Cabinet sub-committee on airports is due to meet shortly to examine the arguments for the three options set out in last year's Airports Commission report.
The letter also confirmed that next week's choice between Heathrow and Gatwick as the Government's preferred option will be followed by a "full and fair" public consultation before a final decision is made.
Successive governments have delayed the decision for years amid fears of a political backlash over moves to expand either Heathrow or Gatwick.
And he revealed that Prestwick Airport could be set for a major boost if the United Kingdom government gives the plans the green light, ahead of a rival expansion scheme at the capital's Gatwick Airport.
However, following next week's meeting, ministers would be allowed to make their views known, albeit after securing Brexit-engaged May's permission.
A decision on Heathrow had already been delayed until after the 2015 election when Cameron appointed the Airports Commission led by former Financial Services Authority Chairman Howard Davies.
The capacity debate has dogged United Kingdom governments for years, with David Cameron, May's predecessor, avoiding a decision first by appointing the commission and then seeking further input on the levels of noise on pollution likely to result from the competing proposals.
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(Adds details, quotes) By Kylie MacLellan LONDON, Oct 18 (Reuters) - Britain will make a long-awaited decision next week on where to allow airport expansion near London, Prime Minister Theresa May's spokeswoman said on Tuesday, with ministers who oppose it given the rare opportunity to publicly voice their dissent.
Tory MP Adam Afriyie said he would vote against Heathrow expansion in defiance of party whips if there was no free vote. "What was once seen as a hard political decision is becoming a much easier one".
A "range of views" was expressed in Tuesday morning's discussion at 10 Downing Street, in which a majority of Cabinet ministers took part, said the spokeswoman.
Mrs May said: "This is a decision of both strategic and national importance".
A decision could then be taken next week by the Economic Affairs (Transport) sub-committee, which is chaired by the prime minister.
Heathrow has always been considered the favourite for expansion.
Both Heathrow, owned by Spanish infrastructure company Ferrovial, Qatar Holding, China Investment Corp plus others, and Gatwick, owned by investment fund Global Infrastructure Partners, Abu Dhabi Investment Authority and others, say that airport expansion can be funded without government money.