But former Conservative leader William Hague warned party rebels that voting against the Prime Minister's final deal could result in Brexit being indefinitely delayed or halted altogether.
International Trade Secretary Liam Fox said the Chequers agreement was created to be a "credible offer" to Brussels to allow negotiations to make progress.
"There is no stronger alliance than that of our special relationship with the USA and there will be no alliance more important in the years ahead", the British leader said.
May is holding a meeting of her new-look cabinet on Tuesday, following a forced reshuffle in the aftermath of the resignations.
He told the programme: "What we are trying to do is not open revolt but we are trying to tell the prime minister and the Cabinet that we have got real concerns about where this is going".
May replaced Johnson with a loyalist, former Health Secretary Jeremy Hunt, and gave Davis' job to Dominic Raab in a bid to shore up her authority.
This could mean Brexit being delayed, a change of government or a second referendum, he said.
There appears to be no immediate challenge to Ms May's leadership, as the Brexit hardliners simply do not have the numbers, and her Conservative Party seems set to weather this storm, despite deep divisions on the issue.
"In a vote of confidence in the House of Commons I would support the Prime Minister that's because I don't want to have another General Election, there isn't a vote of confidence in either case at the moment, I don't think many people want another election, we only had one a year ago".
Mr Buckland added that there had been a realisation that "we all hang together or we all hang separately".
Bradley said that the Brexit plan agreed by the Cabinet last week at Chequers would damage the UK's opportunities to develop global trade and be "an outward-looking nation in control of our own destiny" following Brexit.
"And it would be a split coming from the top, not from the members of the Conservative Party across the country".
"Because I've already cleared the existing text with Mrs Merkel". "That was not the intention of David Davis or Boris Johnson in resigning, but it may yet be their legacy".
Unhappily for May, his most effusive praise was reserved for the outgoing foreign secretary.
This has focused on whether the United Kingdom should prioritise business interests and keep a close relationship with the European Union in order to maintain trade links - which critics say will mean the United Kingdom still abiding by European Union rules and leaving "in name only".
Meanwhile, Justice Secretary David Gauke warned Tory Eurosceptics that a "no deal" Brexit was "not an attractive option at all".
While insisting that the option cannot be ruled out, Mr Gauke told Today: "What I would say to those of my colleagues, if there are some, who think this is pain-free and this is just something that we can ride over very easily, is no deal will have a negative impact on our constituents, on the British public".
"It may resolve the dilemma the prime minister faces".