The United Kingdom leaves the EU's orbit on Dec. 31, when a transition period of informal membership ends following its formal departure last January, and the sides are trying to secure a deal to govern nearly $1 trillion in annual trade.
Prime Minister Boris Johnson, who is also tackling Europe's worst official death toll from COVID-19, says a deal would be preferable but that Britain, which joined the EU in 1973, would flourish without one.
Talks in London over the weekend were "quite difficult" and "massive divergences" remained on elements of fisheries, economic fair play and settling disputes, an EU source said.
"We are running out of time here," Irish Foreign Minister Simon Coveney said.
Echoing his remarks, German Chancellor Angela Merkel said some EU member states were losing patience.
"We don't need a deal at any price and we have made this clear... A deal is in everyone's interest," she said.
Johnson's spokesman said there had been some progress but "there still remains divergence on issues (such as) fisheries and the level playing field."
"We want to try and reach a free trade agreement as soon as possible but we've been clear we won't change our negotiating position," the spokesman said.
A trade deal would not only safeguard trade but also buttress peace in British-ruled Northern Ireland, though some disruption is almost certain at the busiest EU-UK border points.
Failure to secure a deal would snarl borders, spook financial markets and disrupt delicate supply chains that stretch across Europe and beyond -- just as the world grapples with the vast economic cost of the COVID-19 outbreak.
Talks between EU chief negotiator Michel Barnier and British chief negotiator David Frost continued on Monday. The EU team were expected to stay in London for two or three more days.
Asked whether there was reason for optimism, Barnier told reporters: "There are reasons for determination."
Coveney told Ireland's Newstalk Radio that a failure to agree on fishing rights could wreck a deal.
"If there isn't an agreement on this, the whole thing could fall on the back of it and that's the worry," he said.
Fishing alone contributed just 0.03% of British economic output in 2019, but it is an emotive subject as many Brexit supporters see it as a symbol of the regained sovereignty they hope leaving the EU will bring. Combined with fish and shellfish processing, the sector makes up 0.1% of Britain's GDP.
Britain wants "zonal attachment" to agree a total allowable catch for its waters - giving it a much larger quota share than if the fish maths were worked out on the EU's proposals.
EU sources said Barnier had offered Britain a 15-18% bigger catch after Jan. 1 but London had demanded 80% and annual quota talks, which the bloc rejects.
(Reporting by Guy Faulconbridge, Kate Holton and Elizabeth Piper in London; Conor Humphries in Dublin and Gabriela Baczynska, Francesco Guarascio and John Chalmers in Brussels; editing by William Maclean, Larry King and Timothy Heritage)
By Gabriela Baczynska and Elizabeth Piper