During a phone call on Tuesday evening, the Prime Minister told Mr Juncker that nothing short of reopening negotiations and removing the Irish backstop would be good enough for his Government to consider signing an exit agreement.
A Downing Street spokeswoman said: "The Prime Minister set out that the UK will be leaving the EU on October 31, whatever the circumstances, and that we absolutely want to do so with a deal.
"The PM was also clear however that unless the Withdrawal Agreement is reopened and the backstop abolished there is no prospect of that deal."
A spokesman for the commission said Mr Juncker used the exchange to repeat his willingness to "work constructively" to look at "concrete proposals he may have, as long as they are compatible with the Withdrawal Agreement".
Mr Juncker said the EU was "fully prepared for a no-deal scenario" but added that the bloc would do "everything it can to avoid such a situation".
"A no-deal scenario will only ever be the UK's decision, not the EU's," he told Mr Johnson.
The backstop was a joint arrangement agreed between the UK and the EU in a bid to avoid border checks in Northern Ireland after Brexit. Former PM Theresa May signed Britain up to a number of Brussels rules to ensure an open border, until a friction-free solution could be decided.
During the discussion with Mr Juncker, the first the pair have had since July 25, Mr Johnson repeated what he told German Chancellor Angela Merkel and French President Emmanuel Macron during his visits to Berlin and Paris last week, by declaring that the UK would "never place infrastructure, checks or controls at the border" in Ireland.
The move effectively creates a stand-off between the two parties, with Mr Johnson testing the EU over whether it would demand Irish border checks to protect the integrity of goods in the European single market after the October 31 Brexit deadline.
Mr Juncker told the Conservative Party leader that the EU 27's support for Ireland was "steadfast" and they would "continue to be very attentive to Ireland's interests".
Mr Johnson is set to send his EU "sherpa" - his Europe adviser David Frost - for talks with Brussels officials on Wednesday to discuss the backstop further.
The move comes after Mr Johnson said on Monday he was "marginally more optimistic" about the prospect of a deal following meetings with key players including Mrs Merkel, Mr Macron and European Council president Donald Tusk.
But there was still a "substantial disagreement" with Brussels over the terms of the existing deal, which has been rejected by MPs.
British officials believe there has been a "rhetorical shift" in recent days from the EU side, who have previously viewed the Withdrawal Agreement and backstop as "sacrosanct".
While sources do not yet believe there has been a substantial change in the EU's position, the UK wants to use the perceived shift to prise open space for negotiation.
Mr Johnson is due to meet Irish counterpart Leo Varadkar in early September for talks which are viewed as crucial if there is to be any movement on the backstop.
A Downing Street spokeswoman said Mr Johnson believed there would be enough time to achieve a revised deal before October 31.
"The Prime Minister has had good discussions with EU leaders both last week in France and in Germany, and at the weekend," the spokeswoman said.
"There is a clear understanding now that the commitment to leave on October 31 is absolute but that the PM wants to do so with a deal.
"He has also been setting out very clearly the reasons why the backstop is unacceptable to us.
"What is clear is that if there is goodwill on both sides, and a determination to get things done, solutions to the backstop exist and the PM believes they should be discussed."