He said it was unclear if there would be enough support to topple May. Still, the Sunday Times, citing Tories it didn't identify, said half her cabinet want her to stand down in the next two years.
While professing loyalty, his interventions have been seen as undermining Ms May and causing unnecessary unrest ahead of the party's conference last week that culminated in a disastrous speech by the Prime Minister, marred by a coughing fit and letters falling off the slogan on the set behind her.
"I think to ordinary voters it's what's inside the speech that matters".
It comes after a former minister said as many as 30 MPs were prepared to demand an immediate leadership contest in order to oust Mrs May. He maintained the MPs supporting him were both Remain and Brexit supporters. The environment secretary said: "No one is burying their heads in the sand".
Scottish Conservative leader Ruth Davidson urged the party to "settle down " while Commons leader Andrea Leadsom ordered the plot's ringleader Grant Shapps to "shut up".
She told reporters: "Now what the country needs is calm leadership, and that's what I am providing with the full support of my Cabinet".
The Observer newspaper said unnamed senior Conservative figures said while May had no long-term future they were pressing her to shake up her team of ministers, hoping new blood would re-energise the party as well as frustrate Johnson's ambitions.
Speaking to Good Morning Britain he said the plan had been to approach Mrs May in private with a list of names to "save that embarrassment" of a formal leadership challenge.
"I think the conference and the lead-up through the summer has shown that that's not going to happen".
Video Rees Mogg and Sir Bill respond to PM's Brexit statement
"Number 10 must be delighted to learn that it is Grant Shapps leading this alleged coup", he told the BBC. "I mean it really is utter nonsense".
"What you are seeing here is probably the coalition of disappointed people who think their brilliant political talents have not been fully recognised".
A leadership contest would only be triggered if Mrs May lost that vote, or chose to quit.
Mrs May added that she had an uncomfortable time during her party conference speech but never considered leaving the stage early.
Her survival has so far been dependent on the absence of an obvious successor who could unite the split party around Brexit and fear of an election that many Conservatives think would let opposition Labour leader Jeremy Corbyn into power.
"I have to say I've not got much time for them".
She said: "I didn't come into politics for an easy life".
"I think there's an very bad lot of people in our party that need to settle down". I'm actually getting a bit narked and I'm sure there's quite a lot of women out there that are (getting) a bit narked at how many male commentators are talking about male cabinet ministers deciding what she should or shouldn't do as if she has no agency in this herself.