The government has ruled out extending the Brexit process into 2021 amid calls for it to be delayed due to the Covid crisis and the deadlock in trade talks.
SNP leader Nicola Sturgeon and London mayor Sadiq Kahn are among those calling for the Brexit transition period to stretch beyond 31 December.
Downing Street said it would “explore every route” to a deal but time “was in very short supply”.
Talks continue in Brussels with 10 days left until the UK leaves EU rules.
The UK has been following EU regulations since it left the bloc on 31 January, but this “transition period” will stop on 31 December, when the UK exits the EU’s internal market and customs union.
The impasse in trade talks, and the disruption at UK borders due to the Covid crisis, has led to renewed calls for the UK to reconsider extending the transition period to allow more time for a deal.
But the prime minister’s spokesman said the UK continued to “explore every route to a deal that’s in line with our principles”.
He added that time was “in very short supply”.
Scottish First Minister Nicola Sturgeon said the spread of a new Covid variant “demands our 100% attention” and suggested it would be “unconscionable” to compound the UK’s problems by not agreeing a trade deal.
Simon Hoare, Conservative MP for North Dorset, called for the “clock to be temporarily stopped” on the talks, to allow time for Parliament to scrutinise any agreement before the end of the year.
London’s Labour Mayor Sadiq Khan also urged the PM to extend the deadline, saying: “With the virus spreading rapidly and our hospitals increasingly stretched, the only thing the country should be concentrating on is fighting the virus.”
But Labour leader Sir Keir Starmer rejected calls to extend the transition period, saying “further dithering” over Brexit would not help the country.
“I think that it would be far better for the government to get a deal over the line, either today, tomorrow or certainly next week,” he said at an event in London.
The government has long ruled out any extension to the Brexit process, insisting that the end of the transition period is set out in law.
Asked whether it might now reconsider, Transport Secretary Grant Shapps told BBC Breakfast “absolutely not”, saying the date had been known about for a “long time” and changing it would “add fuel to the fire”.
He said it was important that people were ready for the substantial changes to the UK’s trading relationship with the EU happening on 1 January whether there is a deal or not.
Members of the European Parliament have been meeting to discuss the situation, after warning time had run out for it to ratify a deal by 31 December.
One potential option, should the two sides reach a deal in the coming days, would be for the European Parliament to approve it in principle by 31 December, before completing the formal ratification early next year.
What happens next with Brexit?
If this happened, short-term measures could potentially be put in place to minimise disruption to cross-Channel trade before new legally binding rules come into force.
If no UK-EU deal is done, both sides will rely on World Trade Organization rules to govern exports and imports. Tariffs could be introduced on goods being, potentially affecting prices.
BBC Europe editor Katya Adler said the two sides had agreed on the vast majority of issues and that disputes over fishing rights remained the main stumbling block.
Prime Minister Boris Johnson has promised a return of sovereignty over UK waters, while the EU insists member states’ fleets must retain some access.