Two EU sources told Sky News that Britain and the EU had made “significant progress” in the Brexit talks on the right of the European navy to fish in UK waters.
They added that there is now a “landing zone” that includes periodic introductions and specific quotas for different species.
However, they said it was not final.
A UK government source told Sky News that “no progress has been made in fishing” and added: “Nothing new has been achieved today.”
Adam Parsons, Sky’s Europe correspondent, said: “What EU sources are saying is that fishing is not their biggest concern now – they think it can be done.
“The biggest hurdle in negotiating is the level playground competition rules.
“This is consistent with what we’re reporting on all weekends.”
The “Level Playing Field” rule raises issues such as state support for business.
At the end of the month, just before the end of the Brexit transition, the UK’s chief negotiator, David Frost, resumed talks in Brussels with his EU counterpart, Michelle Bernier, to try to resolve the “significant difference”.
It came after an hour-long discussion between Prime Minister Boris Johnson and European Commission Ursula von der Leyen on Saturday evening.
The two leaders agreed on the ultimate pressure to reach an agreement.
In a joint statement after their discussion, Mr. Johnson and Mrs. Von der Lane said that “progress has been made in many areas … there are important differences on three important issues: stratified playgrounds, administration and fisheries.”
British sources later warned that the process could end without a deal.
A source close to the discussion said: “This is the final throw of the side.”
Environment Secretary George Eustace On Sunday, the European Union was accused of making “ridiculous” claims about future fish rights.
He said that while the UK was ready to propose a multi-year agreement for up to three years, the European Union insisted on “permanent” access to British waters.
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Mr Utsai told Sky News: “The EU has suggested a very modest increase that they would tolerate fishing in the UK’s own waters, but we now have only half the fish in our own waters that would not be possible.
“There is also the more important principle that you will not be able to provide any guarantee for the long term, otherwise you are effectively guaranteeing access to the stability of our waters which is not right under international law.
“We can be the only country in the world that can agree to this, so this kind of advice is really quite ridiculous and not in line with international law.”