He broke the news when asked if he feared he was going to be sacked.
He is expected to be a prominent opponent of no-deal from the backbenches.
May will step aside when the victor of the ongoing Conservative leadership contest is announced on Tuesday, bringing to an end a three-year tenure that will be defined by her inability to take Britain out of the EU.
Both candidates have promised to renegotiate the withdrawal agreement with the European Union and its most controversial part, the backstop protocol - an insurance mechanism which aims at keeping an open border in the island of Ireland. The Post notes that Johnson, who began his career as a journalist, "has a loose relationship with the truth".
"I think if the text isn't changed in any way, then it is hard to envisage how that will go through the House of Commons". Britain looks set for a fall showdown between the new Conservative government and British lawmakers determined to thwart a no-deal exit. "This is an anti-Johnson march, which means it's anti-Brexit".
He said: "The figure you quote relates to a Treasury forecast for 2035 - a full 15 years after the end of the implementation period - and it also assumes that the Government would take absolutely no action in response to a no-deal".
The "No to Boris, yes to Europe" protest saw a giant inflatable modelled on the leadership candidate - inspired by the Trump baby blimp - fly in Parliament Square on Saturday.
But halting a prime minister who is determined to press for a no-deal Brexit could be hard.
He added that he did not believe his former colleague could win a deal with the European Union by the deadline.
"I think ordinary people have been targeted and after Brexit the poor will become poorer".
But he seemed to lack the moves to put Mr Johnson off his pace, as the two tangoed with destiny in the battle for the Tory crown.
Brexit has reignited the Scottish National Party's push for independence. Newly elected European Commission President Ursula von der Leyen says Britain can have an extension if it's for a strong reason such as an election or a new referendum.
He has also refused to rule out voting against the Government in a confidence motion - if the next Prime Minister proceeds taking the United Kingdom out of the European Union without an agreement in place.
One Tory rebel suggested that the Labour Party is likely to call a no-confidence vote in a bid to stop a no-deal Brexit: "We just want to keep Johnson and the ERG sweating over what we will do but if no-confidence is the only way to stop no-deal then many of us could support it".
A senior source said: 'Boris appreciates the fact that while the rest of the Cabinet sat on their hands over the deal, David did the honourable thing'.