Talks between the UK and EU are due to resume later in a final bid to agree a post-Brexit trade deal.
After a weekend of tense negotiations, EU sources told the BBC an agreement on fishing was close - but this was disputed by Downing Street.
The UK's chief Brexit negotiator, Lord Frost, will continue talks with his EU counterpart, Michel Barnier.
And PM Boris Johnson will speak on the phone later to European Commission president Ursula von der Leyen.
Border checks and taxes will be introduced for goods travelling between the UK and the EU if a trade deal is not reached and ratified by the end of the year.
The British Chambers of Commerce has warned traders are unprepared for changes that will come when the Brexit transition period ends in 24 days' time.
Mr Johnson and Mrs von der Leyen decided to resume negotiations after trade talks were paused on Friday due to "significant differences" over three "critical issues".
In a joint statement they said fishing rights, competition rules and how to enforce agreed rules were still causing problems, adding that a deal would not be "feasible if these issues are not resolved".
EU sources suggested on Sunday evening that a breakthrough in talks on post-Brexit fishing rules was close to being made.
Brussels wants EU fishermen to have ongoing access to UK waters after Brexit, and the two sides have been discussing a transition period where access could be reduced over time.
One senior EU diplomat told the BBC that fishing "was definitely not the issue the talks are stuck on".
But this was quickly followed by a No 10 source saying there had been "no breakthrough", adding: "Nothing new has been achieved on this."
Both sides are clear that disputes remain on the two other sticking points - the so-called "level playing field", and governance.
What is known as the level playing field relates to a set of shared rules and standards to ensure businesses in one country do not have an unfair advantage over their competitors in others.
Brussels wants the UK to adhere to EU rules on issues like workers' rights, environmental regulations and state aid, but the UK says the goal of Brexit was to break free from following common rules.
BDST: 1201 HRS, DEC 07, 2020