The government has ruled out extending the deadline for reaching a post-Brexit trade deal into 2021, amid a deadlock in talks and a growing Covid crisis.
SNP leader Nicola Sturgeon and London mayor Sadiq Khan want the UK to follow EU trading rules beyond 31 December to allow more time for an agreement.
But Downing Street said it would “explore all routes” to a deal before then, despite time being “short”.
UK-EU talks continue, with 10 days left to reach and ratify any agreement.
The UK has kept following EU regulations since it left the bloc on 31 January, but it will exit its internal market and customs union when this “transition” period finishes at the end of the year.
Without a trade deal, both sides could place import taxes on each other’s goods, potentially affecting prices.
Ms Sturgeon, Scotland’s First Minister, said the spread of a new Covid variant – which has led to more than 40 countries banning people travelling from the UK – “demands our 100% attention”.
It would be “unconscionable” to compound the UK’s problems by not leaving more time to agree a trade deal, she added.
And Labour’s Mr Khan urged Boris Johnson to extend the deadline, saying the UK “should be concentrating on… fighting the virus”.
Fishing rights are a major point of contention in UK-EU trade talks
But the prime minister’s spokesman said the UK would continue to “explore every route to a deal that’s in line with our principles”.
He acknowledged that time was “in very short supply”, saying: “We will need to ratify any agreement ahead of 1 January. The leader of the House [of Commons] made clear that we would recall Parliament in order to give MPs a vote on the necessary legislation.”
Labour leader Sir Keir Starmer also rejected calls to extend the transition period, saying “further dithering” 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.
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 reconsider, Transport Secretary Grant Shapps told BBC Breakfast this was “absolutely not” going to happen.
He said it was important businesses prepared 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 agreement soon, would be for the European Parliament to approve it in principle by 31 December and complete formal ratification early next year.
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.
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.