Theresa May has said she is “pretty resilient” and it is not her style to “hide from a challenge” despite a mishap-strewn party conference speech.
The prime minister told the Sunday Times: “I am a very determined person. I am not someone who gives up.”
But Downing St says the newspaper’s report that she is planning a cabinet reshuffle is “speculation”.
On Friday, former party chairman Grant Shapps claimed 30 Tory MPs backed his calls for a leadership contest.
It followed a conference speech in which Mrs May apologised for calling a snap general election and losing the Tory majority, only to then be plagued by a persistent cough.
She was also interrupted by a prankster giving her a fake P45 and letters falling off the Conservative message in the background.
After Mr Shapps was publicly named as the Tory gathering support for a leadership contest, Mrs May insisted she was providing “calm leadership” with the “full support” of the cabinet.
She then told the Sunday Times the problems during her speech had been “really frustrating”, but added: “Let’s keep this in proportion. I had to give a long speech with a bad cough, a somewhat shaky set and a so-called comedian intent on getting his 15 minutes of fame.
“Was it uncomfortable? Certainly. But let’s not get carried away!”
She added: “The truth is, my feelings can be hurt, like everyone else, but I am pretty resilient.”
Asked what she would do about Foreign Secretary Boris Johnson – who has been seen by some as disloyal after writing two articles setting out his own vision for Brexit – she said she would not “hide from a challenge”, and would “make sure I always have the best people in my cabinet, to make the most of the wealth of talent available in the party”.
But she added: “I have a terrific cabinet.”
Writing in the Sunday Telegraph, Mr Johnson himself suggested only “nutters” in the party would want to oust Mrs May, while in the Mail on Sunday, former Conservative Prime Minister Sir John Major accused “self absorbed” critics of undermining her.
Sir John said the country “has had enough” of the “disloyal behaviour we have witnessed over recent weeks”.
Speaking on BBC One’s Andrew Marr Show, Scotland’s First Minister Nicola Sturgeon said any politician watching Mrs May’s speech would have sympathised with her, but if she wasn’t a “weak prime minister presiding over a deeply divided party” it would not have been an issue.
But the Conservative leader in Scotland, Ruth Davidson, told the BBC that while there had been some “unfortunate shenanigans” in the party, “the pushback has been pretty strong” against those trying to force a leadership contest.
She criticised “tittle-tattle” by colleagues, adding that being a politician is “about delivering for the country, it’s not and should never be about private ambition”.
Of Mr Johnson, whom some Conservative figures have suggested should be sacked, she said: “He is a big intellect, a big figure in the party and if the prime minister believes he is the right person to be foreign secretary then she has my full support.”
Mr Johnson had said he was “fully behind” the PM’s Florence speech last month – designed to break the deadlock in Brexit negotiations – and she should “hold him to that”, Ms Davidson added.
Of her own leadership ambitions, Ms Davidson said she was focussed on her job as Scottish Tory leader: “I’m looking to 2021 [Holyrood elections] and I’m not looking past it because there is quite a lot in my in-tray right now.”
The Conservative former deputy PM Lord Heseltine said he believed a reshuffle was now inevitable given the prime minister’s “unenviable” position – but it would be dangerous to return Mr Johnson to the backbenches.
“Brexit is the over-arching issue of our day and it is hugely damaging to the unity of the Conservative Party,” he told the BBC.
“They can’t make up their minds, either about the issues or the personality they would like to put in her [Theresa May’s] place and that’s the argument for the reshuffle because it could broaden the choice.”
To trigger a vote of confidence in the party leader, 48 of the 316 Conservative MPs would need to write to the chairman of the backbench 1922 committee.
A leadership contest would then only be triggered if Mrs May lost that vote, or chose to quit.