Bridgen and Dorries are both consistently outspoken, and No 10 will be unsurprised at their stance, but Bradley and Caulfield were not previously regarded as troublemakers. She was expected to give a news conference alongside May.
Hunt said it was a "massive wrench" for him to leave the department, adding: "I know some staff haven't found me the easiest Health Sec".
"It would not have been a plausible thing to do and I wouldn't have done a good job at it".
However, she has support from moderates and there has been no formal challenge to her leadership so far.
On Monday, high-profile Brexiteer and Conservative MP Johnson announced he was resigning from the Cabinet, following the agreement of a Brexit proposal that would see the United Kingdom retain close links to the European Union after exiting the organisation.
Conservative party leadership rules mean rival contenders for the top job can not simply challenge the prime minister.
Mrs May was forced to reshuffle roles in her cabinet following a series of resignations, including David Davis and Boris Johnson after the Priem Minister had agreed to a position on her Brexit model at Chequers last Friday.
"I think it is right that the Cabinet backs the prime minister and speaks with one voice and if people don't do that then they have to go", Justice Secretary David Gauke told BBC radio.
The latest chaos for May unfolded within 48 hours of what was supposed to be a breakthrough in uniting a fractious government behind her road map for Brexit. These have all but stalled because of her reticence of show her cards until now for fear of angering one of the two main factions of her party.
But the furore has gained the attention of her allies.
Asked Tuesday whether May should be replaced as prime minister, Trump said it was "up to the people, not up to me".
May's personal authority was badly damaged after she called a snap election previous year to strengthen her hand in Brexit talks but instead she lost her parliamentary majority, meaning she has to rely on a small Northern Irish party to govern.
In response, May said she was looking forward to meeting Trump when he arrives on Thursday: "We'll be talking positively about how we can continue to work together in our special relationship".
The EU definitely expects more concessions from Theresa May in negotiations if she wants what she describes as a deep and special partnership post Brexit.
Earlier, Downing Street announced Johnson's resignation as foreign secretary, just minutes before May addressed MPs about the softer Brexit plan agreed at Chequers on Friday.
Creating a political storm when he compared European Union efforts to create a superstate to Hitler's attempt to rule the continent, saying: "Napoleon, Hitler, various people tried this out, and it ends tragically". "It's not even an accidental betrayal, it was planned and plotted well in advance", said Conservative lawmaker Andrew Bridgen. But should they ultimately do so, the prime minister also looked to have enough support.
May announced on Friday that her warring Cabinet had finally agreed to a plan to follow European Union rules for trade in goods, raising hopes that long-stalled talks with the bloc could progress. But within 48 hours Davis had quit, saying she had already given too much away to the European Union, and Johnson followed. He said "the Brexit dream is dying, suffocated by needless self doubt". They are ones that want to see a "hard Brexit"-Britain crashing out of the EU's single market so long as it can control its borders".
The Sun, Britain's biggest-selling newspaper, said there had been "blunder after blunder" by May.
The New Statesmansays Leadsom is also "in with a shout" for the leadership, but the Daily Mirror points out that her failed bid to beat May in 2016 could damage her chances.