Boris Johnson and Ursula von der Leyen have agreed to continue Brexit talks beyond the Sunday deadline despite a UK official earlier claiming the EU offer was "unacceptable".
Negotiators will continue talks on a post-Brexit trade deal following a phone call between Johnson and the European Commission President on Sunday morning.
It follows late night talks on Saturday between the negotiation teams who are seeking to break the deadlock over state aid provisions and fishing rights.
In a joint statement following the call, they said: "We had a useful phone call this morning. We discussed the major unresolved topics.
"Our negotiating teams have been working day and night over recent days.
“And despite the exhaustion after almost a year of negotiations, despite the fact that deadlines have been missed over and over we think it is responsible at this point to go the extra mile.
“We have accordingly mandated our negotiators to continue the talks and to see whether an agreement can even at this late stage be reached.”
Speaking after the talks, Johnson said the UK "certainly wouldn't be walking away" but claimed the two sides were still "very far apart" on key issues.
"We are always happy to talk and to make progress where we can, I do think as I say there is a deal to be done, if our partners want to do it," he said.
"But we remain very far apart on these key issues. The UK can't be locked in to the EU's regulatory orbit and we have obviously got to take back control of our fisheries...
"Those are points and I think it is very clear what the UK is talking about. Let's see what we can achieve, but in the meantime, get ready, with confidence for 1 January and for trade on WTO terms if we have to."
He added: "The UK should continue to try and I think that is what the people of the country would want me to do.
"We are going to continue to try and we are going to try with all our hearts. We will be creative as we possibly can, but what we can't do is compromise on the fundamental nature of what Brexit is about, and that is being able to control our laws, control our fisheries.
"It is very, very simple, and I think our friends get it. We remain willing to talk and will continue to do so, but in the meantime lets get ready for the WTO option."
Johnson and Von der Leyen had agreed during a dinner on Wednesday that a "firm decision" would be made about the future of the talks today, but remained open to extending negotiations for a further few days if a deal was close.
But Johnson had insisted earlier this week that is was still "very, very likely" that the UK would leave the EU without a deal.
Speaking earlier on Sunday, foreign secretary Dominic Raab had said the high-level talks were crucial to breaking the "political logjam" around fishing and state aid, adding that he couldn't "close the door" on talks continuing beyond the mutual deadline if Brussels indicated it would move on the issues.
"If we're 99% there on the outstanding issues, you wouldn't want to leave any stone un-turned, but I think it's quite a high bar," he added.
The UK's chief negotiator Lord Frost had met his EU counterpart, Michel Barnier, in Brussels on Sunday morning to continue talks, despite a UK government official stating the EU's demands remained "unacceptable".
They said: "The prime minister will leave no stone unturned in this process, but he is absolutely clear: any agreement must be fair and respect the fundamental position that the UK will be a sovereign nation in three weeks' time."
The decision to extend talks come after UK ministers ramped up preparations for no-deal, including ordering Royal Navy ships be put on standby to protect British fishing waters after the end of the transition period on 31 December.
According to The Sunday Times, supermarkets had also been asked to prepare stockpiling foods to ensure there was no shortages if no-deal measures delayed hauliers from importing goods into the UK.
John Johnston is a reporter for Civil Service World's sister title PoliticsHome, where a version of this story first appeared.