Negotiators from the UK and EU are to begin a new push to reach agreement on post-Brexit trade after both sides agreed "to go the extra mile", APA reports citing BBC.
A UK source said the "process still has some legs" but Boris Johnson has warned a no-deal is the "most likely" outcome.
A deadline to finish talks had been set for Sunday, but the prime minister and European Commission president Ursula von der Leyen agreed to an extension.
The pair discussed "major unresolved topics" during a "constructive" call.
They agreed to tell negotiators to carry on talks in Brussels "to see whether an agreement can even at this late stage be reached".
They did not say how long these latest talks would continue, but the ultimate deadline is 31 December, and time must be allowed for the UK and European Parliaments to vote on any deal that emerges before then.
The EU's chief Brexit negotiator, Michel Barnier, will brief ambassadors of the 27 member states in Brussels later.
The major stumbling blocks in negotiations have been over fishing rights, a level playing field for businesses to operate and on how any agreement should be policed.
BBC political correspondent Iain Watson said he was told that the more detailed the talks, the more problems - beyond fishing and competition rules - are beginning to emerge.
The UK and EU have been carrying out negotiations for a post-Brexit trade deal since March and are attempting to secure one before the so-called transition period ends on 31 December - when the two sides would move to trading on World Trade Organization (WTO) rules.
Without a trade deal, tariffs - charges on goods being bought and sold between the two sides - could be introduced and, in turn, prices on certain products may go up.
Reading out a joint statement on Sunday, Mrs von der Leyen said: "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."
Mr Johnson later said "where there is life, there is hope", and that the UK "certainly won't be walking away from the talks".
But he added: "I've got to repeat the most likely thing now is of course that we have to get ready for WTO terms.
"As far as I can see, there are some serious and very difficult issues that currently separate the UK from EU and the best thing to do now for everybody… [is to] get ready to trade on WTO terms."
Labour's Rachel Reeves welcomed the continuation of the talks and said the worst outcome would be to "crash out with no deal whatsoever on 1 January".