Tory grandees are warning David Cameron that he will face a heavy parliamentary defeat if he fails to back a third runway at Heathrow airport, in line with the morning’s recommendation by Sir Howard Davies’s independent airports commission.

The prime minister and the Tory chief whip, Mark Harper, have been told that the pro-Heathrow contingent in parliament – comprising a sizeable number of Tories, the Labour party, the SNP and the DUP – has a clear Commons majority.

“This has absolutely got the prime minister by the nuts,” one senior Tory said. “Unless he is completely shameless - and he is not completely shameless - he can’t possibly overturn the report which has made such a strong recommendation. The numbers don’t add up for anywhere other than Heathrow.”

The warnings to the prime minister were issued after the airports commission, chaired by the economist Sir Howard Davies, strongly endorsed a third runway to the north-west of Heathrow to boost Britain’s long-term airports capacity instead of a second runway at Gatwick.

The Davies report said: “Heathrow offers a stronger solution to the UK’s aviation capacity and connectivity needs than a second runway at Gatwick.”

But the commission said the third runway could only be built if Heathrow met stringent conditions on noise and air pollution. Those conditions should include a ban on night flights, legally binding caps on noise and air quality – and legislation to rule out ever building a fourth runway.

The report said Heathrow’s benefits were significantly greater than Gatwick’s. Davies said Gatwick presented a “plausible case for expansion” but was “unlikely to provide much of the type of capacity which is most urgently required: long-haul destinations in new markets.”

The findings in favour of Heathrow in the commission’s report, which was described by the transport secretary, Patrick McLoughlin, as “clear and reasoned”, create a personal and political headache for the prime minister. McLoughlin told MPs that he would announce the government’s decision in the autumn.

A decision to endorse the report’s findings would mean Cameron having to explain why he was abandoning his “no ifs, no buts” pledge in 2009 to oppose a third runway at Heathrow.

The prime minister would also spark a major cabinet row which could see the resignation of Justine Greening, the international development secretary, who has fought three general election campaigns in the west London seat of Putney opposing Heathrow expansion. Theresa May and Philip Hammond are also opposed, as is Boris Johnson, who has pledged to stop the bulldozers laying the foundation of the new runway.

The pro-Heathrow Tories say that Davies has helped the prime minister to abandon his “no ifs no buts” pledge because the report makes clear that the third runway examined by the commission is different to – and an improvement on – the proposal which prompted Cameron’s remarks in 2009. A succession of senior Tories, including Kenneth Clarke, Andrew Mitchell and Sir Nicholas Soames, told the government it would be acting in the national interest if it opted for Heathrow.

The Tories were encouraged by McLoughlin’s description of the report as clear. But they said they have more work to do to win over the prime minister.

One pro-Heathrow minister said: “There is a very good argument in favour of Heathrow. It is called delivering the long-term economic plan.” This was the prime minister’s general election campaign slogan.

Soames, the former defence minister and Tory MP for Mid Sussex who is a member of the Gatwick co-ordination group opposed to a second runway at the West Sussex airport, told the Guardian: “It is a brilliant report. The prime minister will find it very, very difficult to go against it. The way in which it is presented is that it is in the national interest.

“Of course a second runway at Gatwick would be viable. Of course it would if you chose to build one. But there is no two hub airport anywhere in the world that is a success. Why would we try and shoot ourselves in the foot when it is such an important industry to us.”

The pressure on the prime minister was shown when Harriet Harman and the SNP signalled their parties’ support for Heathrow. Harman teased the prime minister for being intimidated by the London mayor as she highlighted suggestions Downing Street was wary of endorsing the report.

“It looks like the prime minister has been overruled by [Boris Johnson],” Harman said. “He should tell him that he is not the leader of the Tory party yet. Will the prime minister stand up for Britain’s interests or will he just be bullied by Boris?”

Johnson said the Davies report should be “filed vertically” and said that expansion at Heathrow was “never going to happen”.

Johnson said: “I just don’t think you can do this kind of thing in 21st century western European society. Yes maybe in some communist regime in the 1950s. But this is too environmentally damaging.”

Zac Goldsmith, the MP for Richmond Park who is the frontrunner for the Tory nomination to succeed Johnson as London mayor, said: “I made a pledge years ago that I would trigger a byelection if my party gives the green light to expansion. Obviously that pledge still stands.”