It is widely expected that Labour will bring a no-confidence motion in the government after Parliament returns to work next month to try to stop a no-deal Brexit, but Mr Johnson's top aide was reported as saying it would not work.
"We want a deal".
Health secretary Matt Hancock has told the BBC that Parliament will be unable to block a no-deal Brexit on October 31 after failing to do so in parliamentary votes back in June.
The government source said this was not the case, but the European Union had to understand that Johnson could not again bring forward the deal that was rejected by Britain's parliament three times, leading to the resignation of his predecessor Theresa May.
British Prime Minister Boris Johnson says he will pump an additional 1.8 billion pounds ($2.2 billion) into the National Health Service, feeding speculation he wants an early election to bolster his one-seat majority in Parliament and support for his Brexit plans.
"Prime Minister Boris Johnson unveiled an additional £1.bn cash pot for the NHS".
"[The Brexit Party] has discovered that there are people out there who are fed up with having been treated as if they weren't important, that there are areas ignored and left and that neither of the mainstreams parties have been picking up on that almost enough", she said.
"It is better to be prepared for "no deal", because Boris Johnson is credible on this threat", a senior diplomat warned after being briefed on last week's talks.
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One of the EU's Brexit negotiators told the meeting that the G7 summit in France at the end of August could mark the point where it became clear a no-deal Brexit was inevitable.
Former cabinet minister Dominic Grieve branded the newly-appointed Downing Street policy adviser "the master of disinformation" over the claim and insisted MPs could still move to shut down a bid to leave the European Union without an agreement.
The message they are getting from Boris Johnson's team is that the United Kingdom is not going to sign another deal unless it involves getting rid of the backstop. But there is always room for talks and negotiations.
"A no-deal Brexit can be prevented only by legislation, not by a mere expression of parliamentary opinion", constitutional expert Vernon Bogdanor of King's College London, wrote in the Guardian newspaper.
Britain's new Prime Minister is moving quickly to meet that and other Brexit pledges, in a bid to restore some of the trust in politicians that has been eroded in the three years since the referendum that deeply divided the country.
It is the outcome of this vote which has prompted Hancock's comments.
"I do not believe there will be no deal, because Parliament has repeatedly voted against no deal", he explained.