Boris Johnson has refused to back down on the backstop after EU chiefs insisted it would not be scrapped to facilitate a Brexit deal.
The new Prime Minister said in an address to the Commons the proposal needed to be dismissed in order to pass a Withdrawal Agreement.
However, EU Brexit negotiator and European Commission President Jean-Claude Juncker both insisted doing so would not be possible, and backed the deal in its current form.
In response to this, Mr Johnson stood by his belief that "the way to a deal goes by way of the abolition of the backstop".
A Number 10 spokesman said: "The Prime Minister today received a call of congratulation from European Commission President Jean-Claude Juncker. The PM thanked the President for his message.
"On Brexit, the PM reiterated that he wants a deal, and will be energetic in pursuit of finding a way forward, but said the Withdrawal Agreement has been rejected three times by the UK Parliament and will not pass in its current form.
"The PM said that if an agreement is to be reached it must be understood that the way to a deal goes by way of the abolition of the backstop. The PM and the president agreed to stay in contact."
The backstop is an insurance policy contained within the draft EU withdrawal treaty which would have kept Northern Ireland in line with EU regulations to avoid extra checks at the Irish border.
However, it has been frequently criticised as a potential threat to the integrity of the UK and was a longstanding point of contention during Theresa May's premiership.
Earlier on Thursday, Mr Johnson's first full day as PM, Michel Barnier branded the new Tory leader's position on the matter "unacceptable".
He also labelled comments made by Mr Johnson as "combative".
After Mr Johnson made his intentions clear in the House of Commons, Mr Barnier said this was "not within the mandate of the European Council".
This comes as Mr Johnson was also told the current Withdrawal Agreement is the "only" one possible, by President of the European Commission Jean-Claude Juncker.
In a message sent on Thursday, Mr Barnier wrote: "PM Johnson has stated that if an agreement is to be reached it goes by way of eliminating the backstop. This is of course unacceptable and not within the mandate of the European Council."
The negotiator said Brussels would analyse any solutions compatible with the existing Withdrawal Agreement, while reaffirming a commitment to rework the political declaration.
"But as suggested by his rather combative speech, we have to be ready for a situation where he gives priority to the planning for 'no deal', partly to heap pressure on thee unity of the EU27," he wrote.
"I note also the many strong reactions to the speech in the House of Commons. In this context we must follow carefully the further political and economic reactions and developments in the UK following the speech."
The message came shortly before Mr Johnson talked with European Commission President Jean-Claude Juncker in a phone call.
In this, Mr Juncker stated that the EU's position was that "the Withdrawal Agreement is the best and only agreement possible".
A European Commission spokeswoman said: "President Jean-Claude Juncker and Prime Minister Boris Johnson spoke on the phone this afternoon.
"President Juncker congratulated Prime Minister Johnson on his appointment, and reaffirmed his commitment to working together in the best possible way.
"President Juncker listened to what Prime Minister Johnson had to say, reiterating the EU's position that the Withdrawal Agreement is the best and only agreement possible - in line with the European Council guidelines."
They also said it was reiterated that there could be changes to the political declaration adding: "President Juncker also underlined that the Commission remains at the disposal of the United Kingdom to add language to the Political Declaration in line with what the 27 EU leaders recalled when they met in April earlier this year, and to analyse any ideas put forward by the United Kingdom, providing they are compatible with the Withdrawal Agreement."
Previously, Downing Street did not respond directly to Mr Barnier's leaked comments but the Prime Minister's official spokesman said: "It's day one, the PM has said he wants a deal and is going to be energetic in the pursuit of that.
"The fact remains, however, that the Withdrawal Agreement has been rejected three times by this Parliament and is clearly not acceptable to the current UK Parliament."
Mr Johnson earlier expressed confidence in the Commons that he could obtain a fresh agreement by the October 31 deadline and said his predecessor's deal could not be brought back for a vote again.
"Its terms are unacceptable to this Parliament and this country," he said.
He also ruled out any tinkering being able to save the backstop, the emergency measure to prevent a hard border on the island of Ireland in the event of a no-deal.
"A time limit is not enough, if an agreement is to be reached it must be clearly understood that the way to the deal goes by way of the abolition of the backstop," he said.
