"We are not hours away from a deal, what we still need to agree is quite substantial," said the person, speaking on condition of anonymity.
The diplomat said there was still a major gap on fisheries and the EU could not agree with the UK's position on state aid, as well as other guarantees of fair competition.
The person stressed trade negotiators discussing how to overcome the three main sticking points in the talks were "within millimetres" from crossing the EU's red lines, a warning to Britain that the bloc's 27 national capitals might still strike down any agreement if they thought it went too far.
"Economically speaking, the level playing field is very crucial. Politically speaking, fisheries is a very important factor because of the way it plays into domestic politics in France," said the person.
As EU-UK talks to safeguard free trade after Dec. 31 go down the wire, France, Belgium and the Netherlands have warned that no deal would still be less damaging than a bad deal for the bloc.
The person stressed the historic importance of fisheries for northern France communities despite the industry being worth only some half a billion euros a year.
The diplomat said the bloc was not willing to change Barnier's mandate to give him more wiggle room and that the negotiators had one shot at presenting a done deal.
"As long as Barnier stays within his mandate, the deal would not have to go to the leaders before it can be approved," said the person.
The diplomat refused to put a new deadline on the EU-UK talks after many have already been breached. The 27 national EU leaders are due to come together remotely on Dec.10-11 and could approve any deal or prepare further for the impact of a no-deal.
(Reporting by Gabriela Baczynska; Editing by John Chalmers)
By Gabriela Baczynska