The Prime Minister could face a confidence vote if the chairman of the backbench 1922 Committee Sir Graham Brady receives letters from 48 Tory MPs.
It is not known how many he has so far received, with Mr Davies becoming the third to go public, after Andrew Bridgen and Andrea Jenkyns.
Mr Davies’ move comes a day after fellow Eurosceptic Simon Clarke revealed he had withdrawn his own letter after changing his mind about trying to topple the Tory leader.
In a letter to voters in his Shipley constituency, obtained by the Yorkshire Post, Mr Davies said: ‘Politics is all about trust and once it is lost it is impossible to win back.
‘Many people have told me that as a result of this they have lost trust in the PM to properly and fully deliver the referendum result.
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‘It is with much sadness that I have to say that I have also lost trust in her to deliver the referendum result too.’
Mr Davies said that Tories faced losing power to Labour if they failed to deliver the Brexit which the country voted for in the 2016 EU referendum.
He wrote: ‘Failure to keep our promise to the electorate will almost certainly lead to the catastrophe of Jeremy Corbyn becoming Prime Minister and I cannot sit back and allow that to happen.
‘This has not been an easy decision and I have agonised over it, but I know in my heart of hearts it is the right decision.’
If Sir Graham receives letters from 48 MPs – 15% of the parliamentary party – he must call a confidence vote in Mrs May’s leadership.
She would need to win the votes of more than half the Tories’ 316 MPs to survive.
Defeat for the Prime Minister would mean an election to choose a new leader, in which she would not be allowed to stand.
But if she won a confidence vote, she could not be challenged in this way again for another year.
The letter comes the day after Boris Johnson issued a call for Mrs May to tear up her ‘miserable’ plans for close relations with the EU after Brexit and return to the ‘glorious vision’ of Global Britain which she set out last year.
In a highly-charged personal statement to the House of Commons following his resignation as Foreign Secretary, Mr Johnson did not make a direct challenge to Mrs May’s position as Prime Minister and Conservative leader.
But he denounced the plan agreed at Chequers and set out in the PM’s white paper last week as a ‘Brexit in name only’ which would leave the UK in a state of ‘vassalage’.
Accusing the Government of ‘dithering’ over its Brexit negotiations, he said that a ‘fog of self-doubt’ had descended on Mrs May’s stance to EU withdrawal since she first set it out in a speech at Lancaster House last year.