There are "just a few hours left" for the UK and EU to agree a post-Brexit trade deal, says Michel Barnier.
Speaking in the European Parliament on Friday, the EU's chief negotiator said it was "the moment of truth" for the two sides to come to an agreement.
He said there was still a "chance" of a deal, but the "path is very narrow".
Boris Johnson said the UK side was willing to "keep talking", but added: "Things are looking difficult and there is a gap that needs to be bridged."
Talks are resuming later between the two teams in Brussels after the prime minister and European Commission President Ursula von der Leyen spoke on Thursday night.
Mrs von der Leyen said bridging "big differences", particularly on fishing rights, would be "very challenging", while Mr Johnson said a no deal scenario was "very likely" unless the EU position changed "substantially".
Earlier, Mr Barnier met fishing ministers from EU states to discuss the ongoing division over the issue.
Labour leader Sir Keir Starmer accused the government of "dithering over Brexit", calling for the PM to "get this deal done" and "deliver it for the British people".
The UK left the EU on 31 January, but has been following the bloc's trade rules while the two sides negotiate a trade deal.
If one is not agreed by 31 December, they will go on to trade on World Trade Organization (WTO) rules, which could see charges introduced on goods being sold and bought - and could lead to an increase in prices.
Mr Barnier said it was the UK that decided on the deadline and the EU would have been willing to extend the so-called transition period into next year so talks could continue.
"If they should leave with an agreement or without, it is nevertheless the Brits that decided on that deadline," he told the European Parliament.
Mr Johnson has insisted the UK will "prosper" with or without a trade deal with the EU.
The talks taking place in Brussels between Mr Barnier and his UK counterpart, Lord David Frost, are aimed at breaking the deadlock on key issues that remain unresolved.
They include rights to fishing waters from 1 January and what is known as the "level playing field" - where the EU does not want UK businesses to get an unfair advantage by moving away from its rules and standards.
On fishing, Mr Barnier said if the UK wants to use its "sovereignty" over its waters to cut access for EU fisherman, "then the European Union also has to maintain its sovereign right to react or compensate adjusting conditions [to access the] single market".
And on the level playing field, he said there needed to be "fair competition" in place, adding: "If the sovereign United Kingdom would like to depart from those standards, that is their right, but it brings with it consequences when it comes to access to our markets without tariffs or quotas."
Speaking on a visit to Manchester, Mr Johnson said the UK position was "always that we want to keep talking if there is any chance of a deal".
But he called for the EU to "recognise the UK has got to be able to control its own laws - that's what people voted for - and we have also got to be able to control our waters and fishing rights".
The PM added: "No sensible government is going to agree to a treaty that doesn't have those two basic things in it as well as everything else.
"Our door is open, we will keep talking, but I have to say that things are looking difficult and there is a gap that needs to be bridged.
"The UK has done a lot to try and help and we hope our EU friends will see sense and come to the table with something themselves."
Why, you might ask, if the EU's priority in negotiations was to protect the single market, is Brussels allowing the issue of fish to endanger the whole deal?
The level playing field is worth a lot more in monetary and political terms to the bloc, but it sounds quite abstract to voters.
However, fishermen and women losing their jobs, industries dwindling... that would be very visible, very quickly, elevating the importance of fishing rights.
It is in coastal countries where governments fear a public backlash if it's perceived they've sacrificed national fishing communities for a deal with UK.
Although the majority of EU members are not coastal nations, and although everyone in the bloc would love to finally put this deal to bed - for political and financial reasons, as well as being plain fed up with the process - the EU as a whole won't try to force member states to sign up if they are unhappy.
Michel Barnier spoke to EU coastal countries on Friday to try to find a compromise position but, because of the "nothing is agreed until everything is agreed" mantra of negotiations, the EU mood is less optimistic about prospect of imminent breakthrough than it was two days ago.
If a deal is reached between the two sides, it would need to be agreed by parliaments in the UK and EU.
Senior MEPs in the European Parliament have said they will not be "rushed" into signing off a deal on their side, and want to see the text of any agreement by Sunday if they are to approve it by the end of the year.
UK MPs have now finished for the Christmas break, but Cabinet Office Minister Michael Gove said on Thursday that they could be called back to ratify a deal in the coming days, were one to be agreed.
The basics
Brexit happened but rules didn't change at once: The UK left the European Union on 31 January 2020, but leaders needed time to negotiate a deal for life afterwards - they got 11 months.
Talks are happening: The UK and the EU have until 31 December 2020 to agree a trade deal as well as other things, such as fishing rights.
If there is no deal: Border checks and taxes will be introduced for goods travelling between the UK and the EU. But deal or no deal, we will still see changes.