The HuffPost reports that May's allies believe that her unpredictable rival - perhaps the most polarising presence in the party - lacks the numbers to trigger a vote of confidence, and that "even if one was sparked, Johnson may also fail to get enough MPs needed to get his name on the leadership election ballot paper sent to party members".
Mrs May raised the prospect of a Jeremy Corbyn-led government to appeal for Tory unity on Brexit at a meeting of the '22 on Monday night.
His dramatic departure was the second resignation of a Cabinet "big beast" in less than 24 hours, after Brexit Secretary David Davis walked out late on Sunday. While International Development Secretary Penny Morduant immediately welcomed the decision saying Rabb is "highly capable, across the issues, attention to detail, Leave supporter and pragmatist", others said Davis' resignation has plunged Britain into "absolute chaos". But leading pro-Brexit lawmaker Jacob Rees-Mogg said he did not think she would face a leadership challenge.More news: Commonly prescribed heart drug is recalled in 22 countries
"But the popular will, as manifested in surveys of public opinion, suggest that at present about 70 percent of the public judge that the Government is handling the Brexit negotiations badly, and it's been on a declining trend pretty much for the previous year".
Under the Conservative Party's rules, "a confidence vote can not be repeated for another 12 months, so if she survived one, she would have the space to complete the Brexit negotiations", he said.
May replaced Johnson with a loyalist, former health secretary Jeremy Hunt, and gave Davis' job to Dominic Raab in a bid to shore up her authority.
But the leader of the Scottish Conservatives Ruth Davidson said Mrs May was "correct to accept the Foreign Secretary's resignation". "Brussels must not mistakenly conclude, as EU Council chief Donald Tusk hinted, that one more push will destroy Brexit.Brexit must and will happen".
The former home secretary Amber Rudd accused Mr Johnson of "backseat driving" in the Brexit negotiations in September a year ago.More news: Ronaldo signs for Juventus from Real Madrid
And more importantly, where does it leave the Brexit negotiations?
"I am sorry - and a little surprised - to receive (the letter) after the productive discussions we had at Chequers on Friday, and the comprehensive and detailed proposal which we agreed as a cabinet", May said in her reply to Johnson.
Admittedly, Mr Rees-Mogg confirmed that the mood was one of sadness at the European Research Group meeting on Monday once Theresa May had announced her Brexit plans to Parliament.
He says the Brexit "dream is dying, suffocated by needless self-doubt", and that Britain is "headed to the status of a colony", having to swallow European Union writ without "any ability to influence these laws as they are made".
"And it would be a split coming from the top, not from the members of the Conservative Party across the country".More news: BMC beats Sky to capture team time trial
"Nine ministers have now resigned from the two key Brexit departments since Article 50 was triggered".
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