Cabinet minister Mr Cairns said the best person equipped to drive a "bold, ambitious plan" was Mr Johnson.
I am supporting Jeremy Hunt to be the next leader of the Conservative and Unionist Party and our next prime minister.
Boris Johnson would use the £39 billion divorce bill to secure a better deal, refusing to hand over the cash unless improved terms were on offer.
However, EU Chief negotiator Michel Barnier has said that no such renegotiation will be possible.
"Post Brexit, we hope this can be done with a deep and meaningful trade, customs and regulatory agreement", he said. "We can get a better deal".
Britain is mired in its deepest political crisis in decades over how, when and whether it should leave the European Union - a decision that will fall to Mrs May's successor and affect both its future role on the world stage and prosperity for generations to come.
Foreign Secretary Mr Hunt claimed the German Chancellor told him Brussels "would look at any solutions" the United Kingdom puts forward to solve the Northern Irish border issue.
"If we're 95 percent of the way there on October 31, would it really make sense to junk that progress and to say "tell you what, we're leaving without a deal anyway"?" he told the BBC. "We can go for much greater economic growth - and still be the cleanest, greenest society on earth".More news: Columbine High School could be demolished and rebuilt
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"You don't need a magic solution for this, the solution exists". He also confirmed he would be willing to delay Brexit by a few days or weeks if a deal with the European Union was close.
The £39 billion settlement would only be paid when there is "greater clarity" about the way forward, he told the Sunday Times. "I would propose to do that because economically it's right and morally it's right".
Johnson said withholding the £39 billion-pound financial settlement will be a "great solvent and a great lubricant" in forcing the European Union to offer a good deal to the UK.
In my role as co-leader of the One Nation Conservatives I listened last week to each of the main candidates being put through their paces in front of up to 100 members of parliament and Jeremy stands out above everyone. The current deadline is October 31.
Rory Stewart said Mr Johnson's comments on the money and the Irish border were "undignified" and "irresponsible".
The Tory backbencher gave her support for the former Foreign Secretary, who she believes will be "single minded" and "not knocked off course".
Conservative MP Priti Patel.
Mr Javid received a boost with an endorsement from Ruth Davidson, the popular leader of the Scottish Conservatives.