In what The Sun describes as a "barnstorming" speech to a packed 1,500-capacity hall, Johnson claimed to back May - but "made what looked like an unabashed bid to replace her, tearing in to her domestic agenda as well as her plans for Brexit".
Jack Aaron, a delegate from Witney in southern England, praised Johnson's "rousing" speech but said it would be "irresponsible" to try to topple May.
To standing ovations, cheers and laughter, Johnson, May's most powerful critic in her governing party, said her so-called Chequers plan to leave the European Union was a "cheat" that would leave Britain "locked in the tractor beam of Brussels".
But his greatest criticism was reserved for the Prime Minister's Chequers plan to keep the United Kingdom in a free trade area for goods with the European Union after Brexit, which Mr Johnson called an "democratic outrage".
In August, he was criticized for comparing Muslim women who wear face-covering veils to "letter boxes", and last month he compared May's Brexit plan to a "suicide vest".
The attack came at the end of the first day of the Conservative Party's annual conference in Birmingham in which its fault lines over Brexit, already exposed, started to crack open with just weeks to go to settle a withdrawal deal with Brussels.
And he warned: "If we get it wrong - if we bottle Brexit now - believe me, the people of this country will find it hard to forgive".
"We've had our arguments, but now is the time to put them behind us, now is the time to come together, because this is a moment for the optimists", he said.
The BBC's Political Editor Laura Kuenssberg wrote on Twitter: "This is what we would euphemistically call a "wide-ranging speech" - a blatant pitch for leadership by Johnson, right in heart of conference where May meant to be in charge".
"You know we, he and I, are all members of the Conservative and Unionist Party".
"Am I disappointed (that he didn't declare)?" "We need to support the Prime Minister, but we're out to change the policy".
The cross-bench peer, whose speech was watched by the Prime Minister, said: "Business is so important that when I heard a former foreign secretary f-business, it showed him up for the irrelevant and offensive person he really is".
"I want also to congratulate my friend Philip Hammond for predicting that I will never become prime minister, the first treasury forecast in a long time I think to have a distinct ring of truth", he said to prolonged laughter and applause.
While Johnson's Brexit message was greeted with the greatest rapture, his attack on the opposition Labour Party and calls for the Conservatives return to its traditional values of low tax and strong policing were also enthusiastically received.
Mr Johnson, who quit the cabinet in July in protest at Brexit policy, was cheered by activists when he urged Mrs May to drop her plan for future co-operation with the EU.
"Boris always attracts lots of attention".
"Unlike the Prime Minister, I fought for this, I believe in it, I think it's the right thing for our country and I think that what is happening now is, alas, not what people were promised in 2016", he said.
Home Secretary Sajid Javid is preparing to set out the immigration blueprint in a speech to the Conservative Party conference.
Numerous prominent Brexit-backing MPs were front and centre for the speech, including former Brexit secretary David Davis, former Conservative leader Iain Duncan Smith and ex-Cabinet minister Priti Patel.
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