She has already said she has taken steps to ensure she speaks "strongly" this time round.
Tory MP James Duddridge has handed in a letter of no confidence against May, citing the prime minister's refusal to "rip up" her controversial Chequers agreement, and her rival Boris Johnson's "inspirational" speech, as his reasons behind the decision.
"If we cheat the electorate - and Chequers is a cheat - we will escalate the sense of mistrust".
With just six months before Britain leaves the EU, May's precarious position at the helm of her party has been further shaken by criticism of her Chequers plan - named after her country residence - at home and in Brussels.
'We must show everyone in this country that we are that party.
We have to make the arguments for free markets, for open economies, and we have to recognise in doing that, that for some people as I said on the steps of Downing Street back when I first became Prime Minister, they do feel that things haven't been working for everyone in the way that they should.
Questioned again later on BBC Radio 4's Today about how she felt that her former colleague appeared to be mocking her, she replied: "What I feel is that I, and this government, and this party, are getting on with the important job. of getting a good deal for the United Kingdom when we leave the European Union".
Other ministers chimed in, with Justice Minister David Gauke doubting Johnson's credentials for leadership.
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He also said he and his son would do "anything we can" to help Ms Dugdale if necessary.
Ian Livermore, 65, from south Gloucestershire said the speech was "superb" and showed why Johnson should be leader.
In an apparent reference to rumours that the anti-Boris Scottish Tory MPs have code-named their plans "Operation A**e", he added: "The reaction in the hall was electric which even those "Operation A***h**es" can't deny".
"Am I disappointed [that he didn't declare]? Sometimes it comes out, sometimes she hides it away, but she is a really strong woman, a really strong Prime Minister and I really hope that comes across in the speech".
Accusing Mr Johnson of lacking the attention to detail to succeed in "grown-up politics", he dismissed the "super-Canada" Brexit deal favoured by the former foreign secretary as a "fantasy world" plan. "No, because I think we probably can't have the self-indulgence of a leadership challenge".
But Mr Mundell said Mr Johnson was "not an asset" to the party in Scotland.
"It was a great speech, he was optimistic, he talked about Conservative values, and he talked about the opportunities if we do Brexit properly", Richard Tice, co-chairman of campaign group Leave means Leave, said after Johnson's speech.
"Chequers has failed, it's not realistic, it doesn't deliver on the Brexit my constituents vote for".