Mr Johnson is clear favourite to take the crown over his run-off rival, Foreign Secretary Jeremy Hunt.
A Johnson victory could spark more Government resignations.
Ministers opposed to his “do or die” pledge to pull the UK out of the EU on October 31, even if there is no deal in place, could exit before Theresa May formally gives up the premiership on Wednesday afternoon.
Mr Johnson has said he is prepared to borrow more to help out the so-called “squeezed middle” of voters, according to reports.
The ex-foreign secretary is in favour of “fiscal loosening” of tight spending controls imposed by Chancellor Philip Hammond, the Daily Telegraph said.
The newspaper reported that Mr Johnson aims to give tax breaks to people earning less than £80,000 a year if he takes power.
Former Conservative leader William Hague wrote in the Daily Telegraph that Mr Johnson faces the “most immediately daunting challenge on entering No 10 since Churchill” nearly 80 years ago.
He said: “You carry the hopes – even of those of us who didn’t vote for you – that you will defy the odds to keep both country and party intact and successful.
“But you face the most immediately daunting challenge on entering No 10 since Churchill, who you have studied so attentively, stepped over its threshold in 1940.”
Lord Hague added that national security - particularly given recent events in Iran – needed to be of concern, saying: “A British leader needs to be part of binding the West together, or Brexit will one day be seen as the prelude to a more catastrophic disintegration.”
Sir Alan Duncan quit as Foreign Office minister on Monday in protest at Mr Johnson’s expected victory, predicting a “crisis of government” if he becomes PM.
Mr Hammond and Justice Secretary David Gauke have already given notice they will resign rather than serve Mr Johnson.
Mr Gauke told The Times that newly elected Liberal Democrat leader Jo Swinson could take votes from the Tories in the event of a no-deal exit.
He said: “If we were to narrow our support to purely being those in favour of a no-deal Brexit I think we would be significantly out of touch with a lot of people who have traditionally voted Conservative – those who live in London, the home counties, and various other relatively affluent parts of the country.”
Employment minister and Johnson backer Alok Sharma called for Tory unity from ministers returning to the back benches.
He told BBC2’s Newsnight: “I hope what they will do is reflect on the fact that the new prime minister, if it is Boris, will actually have a mandate from the parliamentary party and from the membership.
“What we will do if we have disunity in the party is risk a Corbyn government.”
Arch Brexiteer Nigel Evans, a senior member of the 1922 Committee of backbench Tory MPs, predicted Mr Johnson’s arrival in Downing Street, telling the BBC: “He will be going in there… with at least half a dozen knives already in his back.”
Mrs May will tender her resignation to the Queen after taking Prime Minister’s Questions in the Commons on Wednesday afternoon for the final time, with the new Tory leader set to enter Number 10 soon after.
What will happen on Tuesday?
9.30am: Theresa May will chair a final meeting of her cabinet.
11.45am - 12pm: The result of the Tory leadership contest will be announced, followed by a speech from the winner
The new prime minister will have to govern with a Tory-DUP majority of just two, after Dover MP Charlie Elphicke had the Conservative whip suspended after being charged with sexually assaulting two women.
The Government majority could be further reduced next week if the Tories lose the Brecon and Radnorshire by-election.