The trio of Remain-supporting Cabinet ministers are believed to be preparing to quit after Theresa May holds her final prime minister’s questions in the House of Commons on Wednesday.
Meanwhile, it is thought they could be joined by more than a dozen junior ministers who believe they will be on the way out of the government any way once Mr Johnson takes office.
Mrs May will go to Buckingham Palace after PMQs to see the Queen to formally resign before the winner of the Tory leadership contest goes to see the monarch to ask permission to form the next government.
But by the time Mr Johnson, the overwhelming favourite to be the next PM, arrives at Number 10 his premiership could already be facing its first crisis.
The former foreign secretary will reportedly spend the weekend finalising his plans for a ministerial shake up.
However, a mass resignation by ministers opposed to his pledge to keep No Deal as a Brexit option will hardly paint the picture of stability which the new PM will want to cultivate.
The claims came after Remain-backing MPs led by Mr Hammond inflicted a damaging defeat on the government yesterday.
Rebel MPs successfully agreed an amendment designed to stop Mr Johnson suspending parliament to force through a disorderly Brexit.
Boris Johnson, pictured in Westminster yesterday, could face an immediate crisis if and when he becomes PM next Wednesday
Justice Secretary David Gauke is believed to be preparing to resign immediately before Mr Johnson arrives at Downing Street to take office
Philip Hammond, pictured yesterday at a G7 meeting in Paris, France, is also expected to quit in order to deny Mr Johnson the chance to sack him
Rory Stewart, the former Tory leadership challenger, is expected to join Mr Gauke and Mr Hammond in quitting while reports suggest at least a dozen junior ministers could also resign on Wednesday
Mr Hammond, Mr Gauke and Mr Stewart all defied a three-line whip to abstain on the crunch vote in order to ensure it succeeded.
While the trio of ministers are reportedly preparing to quit on Wednesday, according to The Times, other members of the government who are opposed to Mr Johnson’s Brexit strategy are expected to stay on and wait to be sacked.
Greg Clark, the Business Secretary, will apparently stay in post but has told friends he does not expect Mr Johnson to keep him in the government.
Remain-backing ministers are now entering what is likely to be their final days in power with many of the view that now is the time to take a stand in order to block No Deal.
Margot James quit yesterday as culture minister in order to vote for the crunch amendment which will make it much more difficult for the next PM to prorogue parliament in the run up to the Halloween deadline.
Meanwhile, The Sun reported that more than a dozen ministers are contemplating quitting as Mr Johnson becomes prime minister - assuming he is victorious in his battle against Jeremy Hunt.
Mrs May was accused of having her final moment of weakness yesterday after she failed to sack Tory rebels who helped Remain-backing MPs secure a majority for an anti-No Deal amendment.
Her failure to discipline Mr Hammond in particular prompted fierce criticism after it emerged that he had sent text messages to fellow ministers to urge them to ignore the three-line whip.
Some 17 Tory MPs voted against the government, while 30 abstained. Mr Hammond was among those to have ignored the instruction to vote against the amendment.
It is the first time in memory that a chancellor has directly disobeyed a three-line whip and stayed in post.
The result yesterday afternoon means ministers will be legally obliged to come to the Commons and stage votes at regular intervals in October - making suspending the House to force No Deal all-but impossible.
Mr Johnson has repeatedly refused to rule out proroguing parliament in order to stop MPs launching a last minute attempt to stop No Deal in the days before October 31.
He has pledged to deliver Brexit 'do or die' by the Halloween deadline, with or without a deal.
But many Tory Brexiteer MPs believe the Remainer rebels wasted their time because Mr Johnson would never actually go ahead with such a plan and it was 'never going to happen'.
One senior Conservative MP mocked the 17 Tory rebels and 30 abstainers who aided opposition MPs as they said: 'They are the stupidest bunch of generals I have ever come across.'
The government was trounced by 315 votes to 274 despite imposing the three-line whip - meaning that Tories who failed to oppose the measure should have faced punishment.
But Downing Street was left humiliated after it admitted a swathe of more senior Conservatives who were among the abstainers would not face any punishment.
Mr Hammond boasted after the vote about his failure to back the government as he reignited his war of words with Mr Johnson.
The Chancellor tweeted after the result: 'It should not be controversial to believe that Parliament be allowed to sit, and have a say, during a key period in our country's history.'