Signs have emerged that a post-Brexit trade deal is still possible, but an EU diplomat has said "differences on fisheries remain difficult to bridge".
With just nine days to go until the end of the transition period, Michel Barnier said the talks were "really in a crucial moment" and the bloc was giving a "final push" to try and reach a deal.
But EU diplomatic sources cited by Reuters news agency said Mr Barnier told EU ambassadors in an update on Tuesday afternoon that fisheries remain a stumbling block.
"Progress has been made. Most issues are preliminarily closed or close to being agreed. However, differences on fisheries remain difficult to bridge," one diplomat said.
"Unfortunately, the EU is not moving enough yet to clinch a deal on fisheries."
The diplomat, who spoke under condition of anonymity, added: "The EU will not close its door to the UK and remains ready to negotiate beyond the 1st of January."
Mr Barnier was also said to have described the latest offer on sharing out fish catches from 2021 as "totally unacceptable".
It comes after Reuters reported that Prime Minister Boris Johnson and European Commission President Ursula von der Leyen held a secret call on Monday evening.
The news agency cited sources who said the two leaders talked about fishing and the coronavirus crisis.
It added Brussels now appeared willing to accept a reduction in the value of its catch in UK waters of up to 25% over a period of time from 2021.
Downing Street is not denying the phone call between Mr Johnson and Ms von der Leyen took place but pointed to commitments made previously that they would stay in touch during negotiations.
In mid-December, both sides were loudly talking up the prospect of no-deal.
But after yet more deadlines passed that they said were the points a deal needed to be reached by, public statements from Number 10 and the Berlaymont (the European Commission headquarters) have gone much quieter.
More speculation a deal could be clinched was boosted by a former Number 10 special adviser on Brexit, who wrote an article for Politico laying out how a compromise could be reached on fish in time.
There is just over a week to go until the end of the transition period - the arrangement the UK entered into once it left the EU on 31 January.
Since then, the country has continued to follow most of the bloc's rules while negotiators try to hammer out an agreement.
Talks appear to be going down to the wire, and with parliament broken up for the Christmas holidays, Sky News has been told it could be recalled on 30 December to approve any last-minute agreement.