The reports came after Mr Johnson's campaign aides moved to play down claims that they are considering plans to prorogue Parliament in late October - a move which could prevent MPs stopping a no-deal Brexit.
Lord Anderson warned that if Britain was "driven over the cliff" on October 31 there was no way back up.
The two candidates have vowed to scrap a "backstop" provision in the draft deal that was insisted on by Brussels to keep the Irish border open.
Conservative Party members are presently in the final stages of selecting the next leader of their party, who through the composition of the present British Parliament will become the next Prime Minister when the votes are counted and the succession made next week.
If the victor, to be announced on July 23, sticks to that pledge it makes the chances of a European compromise all but impossible before the Brexit deadline.
Prime Minister Theresa May launched a broadside against populism on Wednesday, using a valedictory speech to encourage her successor to embrace compromise to secure Britain's exit from the European Union.
Now it is clear the victor must also persuade Brussels to drop one of its most steadfast demands - an insurance policy created to prevent the return of a hard border between EU-member Ireland and British province Northern Ireland.
A no-deal Brexit could tip Britain into recession, undermine sluggish euro zone growth and weaken London's claim to be the world's pre-eminent worldwide financial center.
"It has led to what is in effect a form of absolutism - one which believes that if you simply assert your view loud enough and long enough you will get your way in the end", she said.
Asked if he wanted to be Britain's next envoy to Washington, Fox said: "No".
She went on to say that populist movements have "seized the opportunity to capitalize on that vacuum".
The new Tory leader, and Prime Minister, will be announced next Tuesday. "We will do everything to support this measure in the Commons tomorrow", he said. "And especially for a Brexiteer".
'If Parliament were to endorse a no-deal Brexit, as it has not done to date, there can be no democratic argument against it, ' he said.
"They have embraced the politics of division; identifying the enemies to blame for our problems and offering apparently easy answers", she said.
Both contenders last night ruled out reforms to the Irish border backstop to revive Theresa May's thrice-rejected withdrawal deal. The now-invisible frontier between Ireland and the United Kingdom province of Northern Ireland is Britain's only land border with the EU.
"I do not think that prorogation is a tactic that any prime minister would employ", she told LBC.
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