The foreign secretary is under fire for missing a key Commons vote later on whether to approve a new £14bn runway.
Mr Johnson, who is on an overseas trip, told constituents he would continue to oppose the runway and doubted it would be built, the Evening Standard said.
MPs are expected to approve the plan later but a senior Labour figure said it could "block" it if was in power.
The Heathrow expansion is forecast to cost about £14bn, and is likely to lead to hundreds of homes being demolished in the nearby villages of Longford, Harmondsworth and Sipson.
A new runway, which was approved by ministers earlier this month, would increase Heathrow's capacity from 85.5 million passengers to 130 million.
With the government issuing a three-line whip to its MPs - meaning they have been ordered to vote in favour - it is widely expected that the expansion plan will get Commons backing.
Up to 40 of Labour MPs are expected to vote for the plan, which is also backed by leading trade unions.
But the SNP's 35 MPs, who had been expected to vote with the government, could decide to abstain or vote against expansion, which could make the result closer than previously expected.
Opponents of Heathrow expansion have attacked the scheme on environmental, noise and financial grounds, with some making the case for an alternative expansion scheme at Gatwick airport.
Shadow Chancellor John McDonnell, whose Hayes and Harlington constituency is near the airport west of London, said the environmental impact of a bigger Heathrow posed "a threat to the planet".
Asked whether Labour could rethink the project if it was in government, he replied: "My view is we've got to block it because it's so dangerous for climate change."
Responding to the same question, Labour leader Jeremy Corbyn said it would depend on "what stage the whole thing has got to by then".
Transport Secretary Chris Grayling has said he is "cautiously optimistic" that MPs from all MPs parties would back "the biggest transport decision in a generation".
The government has pledged the airport will be built at no cost to the taxpayer, will create 100,000 jobs and will benefit the entire country, through guaranteed internal flights to the rest of the UK.
The new runway, Mr Grayling. Was "not simply a project for London and the south of England" and "the connections we create through Heathrow will benefit the whole of the UK".
Ministers insist the project will have built-in environmental protections, with the ability to fine Heathrow or ground aircraft if promises on night flights and other contentious issues are broken.
'Matter for Boris'
When he was elected MP for Uxbridge and South Ruislip in west London in 2015, Mr Johnson, pledged to lie in front of bulldozers to stop expansion when he was London mayor.
The government has so far declined to say where Mr Johnson, who has long promoted a scheme for a new airport in the Thames estuary, will be when MPs vote on Monday evening, on security grounds.
The Evening Standard quoted a letter it said Mr Johnson had sent to his constituents before leaving the country, in which he defended his behaviour.
"I have long been an opponent of a third runway at Heathrow and that is why I am not voting for it tonight.
"It is clear from what is likely to be a large majority of MPs who are in favour of a third runway that my resignation would have achieved absolutely nothing," he wrote.
"In view of the very considerable difficulties that still face the third runway - its cost and the appalling air and noise pollution entailed by the project - I believe it will be a very long time before we have to make good on that pledge; if indeed a third runway ever comes about."
But former Tory minister Greg Hands, who did quit over the issue last week, said it was "very important" for politicians to fulfil "clear pledges" while Conservative MP Robert Halfon said Mr Johnson's constituents would "quite respect" him if he did quit.
But MP Jacob Rees-Mogg told LBC Mr Johnson's decision to absent himself was "not unreasonable" given the conflict between his longstanding opposition to a new runway and his party's support for one.
Chancellor Philip Hammond, who has also in the past been critical of plans for a new runway, will also miss the vote as he is in India on a trip to promote UK financial services.
The Department for Transport has previously said no expansion would mean London's five airports reach full capacity by 2034.
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