The Prime Minister will travel to Luxembourg this morning for his first face-to-face meeting with Mr Juncker since taking office in July.
A Number 10 source said Mr Johnson “could not be clearer” that he won’t accept further delays to the UK’s withdrawal from the EU.
They added: "Any further extension would be a huge mistake. It is not just a question of the extra dither and delay - it is also the additional long months of rancour and division, and all at huge expense.
“We must finally deliver on the 2016 referendum result.”
Boris Johnson ‘cautiously optimistic’ of new Brexit deal
However, the source also insisted that the PM was working tirelessly to secure a deal by October 18.
Mr Johnson has reinforced his commitment to reaching an agreement with the EU, saying he is working "flat out" to do so.
Writing in the Daily Telegraph on Monday, Mr Johnson said he believes “passionately" that a new Brexit deal can be struck with Brussels, but reiterated his ““do or die” pledge to take the UK out of the bloc next month.
He wrote: "If we can make enough progress in the next few days, I intend to go to that crucial summit on October 17 and finalise an agreement that will protect the interests of business and citizens on both sides of the Channel, and on both sides of the border in Ireland.
"I believe passionately that we can do it, and I believe that such an agreement is in the interests not just of the UK but also of our European friends.
"We have all spent too long on this question. And if we can get that deal, then of course there will be time for Parliament to scrutinise and approve it before the end of October.
"But be in no doubt that if we cannot get a deal - the right deal for both sides - then the UK will come out anyway."
The PM will also hold talks with Luxembourg's prime minister Xavier Bettel and the EU's chief negotiator Michel Barnier on Monday.
Mr Johnson will be accompanied by Brexit Secretary Steve Barclay and fellow Brexit representative David Frost.
Downing Street has sought to downplay speculation that Monday's meeting could be a breakthrough moment, and Mr Barclay said on Sunday that there was still "significant work" to do to reach an agreement.
Brexit Secretary Stephen Barclay on Boris Johnson likening Britain to The Incredible Hulk
However, he said a "landing zone" for a deal was in sight, telling Sky News's Sophy Ridge On Sunday that there had been "extensive talks" at a technical and a political level.
Mr Barclay, in an interview with the BBC, also suggested the transition period could be extended beyond 2020 to resolve issues relating to the backstop.
Over the weekend, Mr Johnson likened Britain leaving the EU to the Incredible Hulk, telling the Mail On Sunday: "Hulk always escaped, no matter how tightly bound in he seemed to be - and that is the case for this country."
But his comparison was described as "infantile" by the European Parliament's Brexit co-ordinator Guy Verhofstadt, who questioned: "Is the EU supposed to be scared by this?"