An exclusive poll of 500 Tory councillors found they would want MPs to back the Prime Minister in any no-confidence vote, by a factor of nearly two to one.
UK Prime Minister said late on Saturday that there is no alternative to the draft agreement on the UK withdrawal from the European Union and urged the UK Parliament to approve the deal in order to avoid "more uncertainty", Sputnik reported.
Labour leader Jeremy Corbyn said he did not believe the text met the needs of the country, while the Democratic Unionist Party (DUP), which is propping up the Conservative Party minority government through a confidence and supply agreement, has vowed to vote down the deal.
"I know that Gibraltar has no intention of being a back door, but instead a centre of high quality financial services", Mr Barclay said at the time.
She is a longstanding ally of May's and has said she supports the proposed divorce agreement with the EU.
They are aiming for the magic number of 48 - the 15 percent of Conservative lawmakers needed to trigger a challenge to her leadership under party rules.
May also said her husband feels the pain of the vicious personal attacks on her by Conservative Party critics even more than she does.
But the Tory grandee Sir Nicholas Soames said: "I am truly dismayed at the dismal behaviour of some of my Colleagues parading their letters to Graham Brady on TV in a vulgar and pathetic display of inferior virtue signalling".
The five ministers were named widely in media reports as Andrea Leadsom, leader of the House of Commons, Environment Secretary Gove, International Trade Secretary Liam Fox, Transport Secretary Chris Grayling, and International Development Secretary Penny Mordaunt.
"A change of leadership at this point isn't going to make the negotiations any easier and it isn't going to change the parliamentary arithmetic", the Prime Minister said.
Seven leading Conservatives, including Boris Johnson, David Davis and Dominic Raab, are actively preparing campaigns to challenge Theresa May's leadership, United Kingdom media reported, as she declared "no alternative" to her Brexit deal.
Four UK ministers holding key posts such as UK's Brexit negotiator Raab handed in their letters of resignation on Thursday citing differences with the draft deal.
"We won't agree the leaving part. until we've got what we want in the future relationship, because these two go together".
The document, coordinated by members of the Alliance of British Entrepreneurs (ABE) and seen by City A.M., asks MPs to be subjective in their assessment of May's proposals, which it believes do not meet the tests set out by the Prime Minister in a speech at Lancaster House past year.
The backstop would mean that Northern Ireland would stay aligned to some European Union rules on things like food products and goods standards, which critics say is unacceptable.