Asked if now was the moment for Johnson to lead the UK, Bannon said: "I believe moments come".
"I will resist - for now - the temptation to bang on about Brexit", Mr Johnson said in a column for the Brexit-supporting Telegraph newspaper, his first public comments since quitting the cabinet a week ago.
However, the danger to the Prime Minister was underlined by the disclosure that Brexiteers had set up a Whatsapp group to co-ordinate voting tactics, organised by ex-Brexit minister Steve Baker, who quit over the Chequers plan.
Mr Davis said "the general direction of policy will leave us in at best a weak negotiating position, and possibly an inescapable one".
The spokesman said Mrs May was "very clear that the proposal we put forward at Chequers delivers on the will of the people in the referendum".
"Let us again aim explicitly for that glorious vision of Lancaster House - a strong, independent self-governing Britain that is genuinely open to the world, not the miserable permanent limbo of Chequers", said Mr Johnson.
Former global development secretary Priti Patel, who is proposing one of four amendments to the Taxation (Cross-border Trade) Bill, told BBC Radio 4's Westminster Hour the white paper had "many flaws around our independence and our ability to make free trade agreements".
She told BBC Radio 4's Today programme: "We simply can not go on like this".
The Government could be forced into a compromise to avoid a damaging blow to the Prime Minister's authority.
Kerber reaches Wimbledon final round, ends Ostapenko's best run
She's here to win it, in only her fourth tournament back after pregnancy, childbirth and a host of complications in the aftermath. The world No 10, Kerber, is the only woman to have reached the second week of every Grand Slam so far this season.
"My job as Prime Minister is to deliver for them, but also I've got to be hard-headed and practical about this and do it in a way that ensures we get the best interests for the United Kingdom".
The Prime Minister faced continued pressure over Brexit from Tory MPs during her final appearance in the Commons chamber for the summer, and will go before the Liaison Committee later.
Writing in The Sunday Times, he said the Cabinet had agreed "concessions to the European Union that were so fundamental they risked undermining the whole Brexit process".
However, a close ally of Philip Hammond, the Remain-backing Chancellor, mockingly described talk of crisis over the Chequers plan as "fake news", adding: "Crisis?"
He warned the Prime Minister not to "make the fatal mistake of underestimating the intelligence of the public, saying one thing to the European Union about what we are really doing and pretending another to the electorate".
May's plans for exiting the European Union were narrowly approved by the House of Commons in a series of votes this week, but only after she bowed to Brexit hardliners led by Jacob Rees-Mogg to salvage her program.
Having delivered his speech, Mr Johnson quickly left the Chamber.
Labour's David Lammy said: "Boris Johnson's after-dinner speaking fee must have plummeted faster than the pound after that dreary, self-important resignation speech".
West Streeting, a Labour MP who supports the campaign for a second referendum, said the speech was "a total damp squib".