- Video report by ITV News Political Correspondent Daniel Hewitt
Both men reported to be being backed by some Cabinet members to replace Theresa May amid speculation that some ministers are plotting a coup to oust her, have said they "100%" back the Prime Minister and do not have ambitions to take over from her.
The support from David Lidington and Michael Gove comes as the Chancellor and a former Conservative Party leader also threw their weight behind Mrs May following reports she has only days left at Number 10 due to a Cabinet coup to oust her.
A number of Cabinet ministers told Sunday newspapers that MPs are plotting to remove Mrs May from Downing Street due to a furious backlash over her handling of Brexit.
Despite the rumours circling around Mrs May's tenure, Political Editor Robert Peston believes Mrs May will not go imminently and that the "coup is over before it even started", but added that the Government is close to collapse due to deep-splits over Brexit.
The Sunday Times reported 11 Cabinet ministers had told the paper they wanted Mrs May to make way for someone else and that the PM's de facto deputy David Lidington was in line to take over the helm as a caretaker prime minister until a leadership contest could be held.
Yet speaking outside his home in Aylesbury, Mr Lidington said he has no desire to take over from Mrs May and said he was "100% behind the Prime Minister".
Meanwhile the Mail on Sunday reported ministers were plotting to install Environment Secretary Michael Gove as a caretaker leader instead.
Yet he too said he "absolutely" supports the Prime Minister, adding: "It's not the time to change the captain of the ship."
- Political Correspondent Daniel Hewitt on why a coup is looking less and less likely, but Theresa May is not out of danger
On Sunday, Mrs May held meetings with Boris Johnson, David Davis, Damian Green, Iain Duncan Smith, Steve Baker, Dominic Raab, Chief Whip Julian Smith, Brandon Lewis, Stephen Barclay, Alistair Burt, Michael Gove, David Lidington and Jacob Rees-Mogg at her country residence, Chequers in a bid to get them to back her Brexit deal.
In a statement released after the meeting, Downing Street said Mrs May held "lengthy talks" with "senior colleagues about delivering Brexit".
"The meeting discussed a range of issues, including whether there is sufficient support in the Commons to bring back a Meaningful Vote this week."
Following the speculation, high profile Tories hit out at what they described as "self-indulgent" actions from those involved when they should instead be focused on Brexit.
Vocal calls for the Prime Minister to go first emerged on Saturday night when Mrs May's former policy adviser MP George Freeman said it was "all over for the PM", tweeting: "She's done her best. But across the country you can see the anger.
"Everyone feels betrayed. Government's gridlocked. Trust in democracy collapsing. This can't go on. We need a new PM who can reach out (and) build some sort of coalition for a PlanB."
Calls for Mrs May to go were echoed in the newspapers, where pro-EU former education secretary Nicky Morgan told the Sunday Telegraph that Cabinet ministers should tell Mrs May "it's time to go" while Brexiteer Steve Baker said potential leadership contenders in the Government should "act now".
Meanwhile Tory backbencher Anne-Marie Trevelyan wrote in the same paper: "We now need a leader who believes in our country and wants to take her on the next stage of her journey."
Conservative peer Lord Gadhia, a former member of David Cameron's inner circle, said the upcoming days in Parliament may be "very dramatic" and could see the end of Mrs May's time as premier.
The wave of discontent against the Prime Minister met with support for her on Sunday, with Mr Lidington saying he was "100% behind the Prime Minister" and had no desire to take over from her.
"I don't think that I've any wish to take over from the PM (who) I think is doing a fantastic job," Mr Lidington said as he spoke outside his Aylesbury home.
"I tell you this: one thing that working closely with the Prime Minister does is cure you completely of any lingering shred of ambition to want to do that task.
"I have absolute admiration for the way she is going about it."
Reports of a Cabinet split and unhappiness with Mrs May's leadership centre on her handling of Brexit and the lack of support for her deal.
Yet Mr Lidington threw his weight behind the Prime minister and her deal, calling it a "really balanced package" which is "in the national interest" and called on both "Leavers and Remainers to get behind it" in order to "deliver an orderly departure from the European Union" and allow further negotiations to take place.
Addressing the rumours of Cabinet members seeking to oust Mrs May, the 62-year-old suggested that this was not the case, instead saying that Cabinet members want to "find ways to get the Prime Minister's deal delivered" whichever side of the Brexit divide they sit on they "support the Prime Minister's deal and we're all behind her in trying to get that through Parliament".
Mr Lidington added that anyone doubting Mrs May's leadership abilities "should look at the enormous amount of work our Prime Minister has put into this [her Brexit deal].
"Whenever I've spoken to her, what I've seen is a woman who's motivated in the national interest - getting the best deal for this country."
Chancellor Philip Hammond likewise threw his support behind the Prime Minister, accusing Cabinet ministers allegedly plotting to oust Mrs May of being "self-indulgent".
Mr Hammond told Sky News he did not think it was the case that Mrs May has run out of road, saying: "This is not about the Prime Minister or any other individual, this is about the future of our country.
"Changing Prime Minister wouldn't help us, changing the party in Government wouldn't help us: we've got to address the question of what type of Brexit is acceptable to Parliament...
"The Prime Minister's deal is my preferred way forward but I'm realistic that we may not be able to get a majority for the Prime Minister's deal and if that is the case then Parliament will have to decide not just what it's against, but what it is for."
While Mr Hammond said changing prime minister would not "solve the problem", he refused to be drawn on whether his colleagues had approached him asking him to make an intervention.
However, he acknowledged that "people are very frustrated and people are desperate to find a way forward in the just over two weeks that we've got to resolved this issue".
Likewise Mr Gove said he "absolutely" supports the Prime Minister, saying: "It's not the time to change the captain of the ship."
"I think this is a time for cool heads. But we absolutely do need to focus on the task at hand, and that's making sure that we get the maximum possible support for the Prime Minister and her deal," he said.
"We need to make sure that colleagues in the Cabinet and across the country recognise that in the course of this week we have very important decisions to make.
"And I hope as many people as possible will recognise that the best way of honouring the referendum mandate and making sure that we leave in an orderly fashion is to support the Prime Minister in her efforts."
He added: "It's not the time to change the captain of the ship, I think what we need to do is to chart the right course."
Also on Sunday, Conservative former leader Iain Duncan Smith lashed out at Cabinet ministers briefing against the Prime Minister in the newspapers, telling BBC One's Andrew Marr Show: "I think that's appalling, I think they should be censured and some of them should be sacked.
"And the idea of a cabal [a secret political faction], a cabal that never wanted to leave the European Union, turning out to decide what should happen over our future would be unacceptable to my colleagues."
Mr Duncan Smith added that any idea of a leadership election would create "complete chaos", and said Cabinet ministers owe Mrs May not to behave as they have.
"I think round the country, in the Conservative Party, and outside the Conservative Party there will be real disgust at the behaviour of some of our Cabinet ministers who are not fit for their positions if they behave like this," he told Andrew Marr.
"They should be apologising and they should shut up for God's sake."
Mr Duncan Smith added that "if there is to be a leadership change that leadership change has to be done through the correct process with the membership out there deciding who will be their leader - not some ghastly five or six man and woman cabal that actually decides things internally."
Mr Duncan Smith also urged fellow Brexiteers to keep their options open on whether to support the Prime Minister's deal if it comes back to the Commons this week.
He said: "I'm going to keep, and I recommend my colleagues do, keep their options open on this because we don't know what's happening this week, we've no idea what the alternatives are and whether people vote for this or not depends hugely on whether we are able to leave with no-deal or not or whether there is a change to this."
Reports of the alleged coup against Mrs May come the day after one million people were said by organisers to have taken part in a march on Parliament demanding a final say for the public over Brexit.
Elsewhere, pro-Brexit campaigners are continuing their long hike from the North East to the capital, leaving Loughborough on Sunday morning.
Sunday marks the end of another turbulent week for the Prime Minister which saw her come under fire for delaying Brexit and seeking to blame MPs for the impasse.
It is thought Parliament will be given the chance to hold indicative votes on alternatives to Mrs May's Brexit deal this week.
Appearing on Sophie Ridge on Sunday, the Chancellor said: "One way or another Parliament is going to have the opportunity this week to decide what it is in favour of, and I hope that it will take that opportunity - if it can't get behind the Prime Minister's deal - to say clearly and unambiguously what it can get behind."
But Mr Hammond said a decision had not yet been made on whether Tories would be given a free vote on the matter.
He also said a second referendum was a "perfectly coherent position", telling the programme: "I'm not sure that there's a majority in Parliament for a second referendum but it's a perfectly coherent proposition - many people will be strongly opposed to it, but it's a coherent proposition and it deserves to be considered along with the other proposals."
Next week it also looks increasingly unlikely that there will be a third meaningful vote on Mrs May's Brexit deal after the Prime Minister wrote to parliamentarians warning if there is insufficient support for her Withdrawal Agreement in the coming days that she could seek an extension to Britain's EU membership beyond the European Parliament elections, potentially leading to a long deal on the UK's EU departure.