But Britain said European Union nationals should be able to travel visa-free to Britain for tourism or "temporary business", and there should be measures allowing young people and students to work and study in Britain.
If Davis's resignation rattled May, Johnson's shook the foundations of her government.
The two resignations leave May badly exposed and raise questions over whether she will stand firm in her commitment to pursue a "business friendly" Brexit, or will be faced with more resignations and calls to quit herself.
"His replacement will be announced shortly".
Asked Tuesday whether May should be replaced as prime minister, Trump said it was "up to the people, not up to me".
The President's comments, made as he prepared to depart for a European trip which will include a visit to the United Kingdom, preceded the resignation of two Conservative vice-chairs over their opposition to Mrs May's plans for Britain's future relations with the EU.
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".
The electorate is totally confused because May has deviated from the message of the 2017 referendum with her soft Brexit landing that she expects to use to placate Michael Barnier, the chief European Union negotiator.
Rees-Mogg told AFP his aim was only "to help the government stick to some of its earlier promises".
Together with a customs partnership (which would require the U.K.to collect tariffs on the EU's behalf), this could remove the need for a physical border between Ireland and Northern Ireland - avoiding one of the greatest dangers aroused by this ill-conceived project - and allow close-to-frictionless trade in goods between Britain and Europe.
Mr Johnson, who was the face of Brexit for many in Britain, said Mrs May was heading for a semi-Brexit.
Theresa May's hold on Downing Street has taken a big hit, but is Boris Johnson's departure enough to oust her?
The appointment of Raab, a leading Brexit supporter, and the decision of some eurosceptic ministers to stay suggest Tory Brexiteers are divided.
Hard-line backbencher Andrew Bridgen sent a letter of no confidence in Mrs May to the chairman of the backbench 1922 Committee, Sir Graham Brady.
Under Conservative Party rules, a confidence vote in a leader can be triggered if 15 percent of Conservative lawmakers - now 48 - write a letter requesting one.
It followed a cabinet meeting at Chequers to get an agreement on the UK's future relationship with the EU. "The Prime Minister can't deliver Brexit and has zero authority left".
Asked if any European Union citizen with a job offer would be able to move to Britain after Brexit, he replied: "That's dependent on what the Government determines of its post-EU migration policy". Euroskeptic Conservative lawmaker Peter Bone said party activists felt "betrayed" by the government plan.