Update: More areas of England will go into Tier 4 from Boxing Day, the health secretary has announced. Read the latest story with the details here.
But Communities Secretary Robert Jenrick has insisted no rule changes will come into force before Christmas Day.
"We are not going to change people’s plans 24, 48 hours ahead of Christmas," he said.
Ministers will meet on Wednesday to decide whether more parts of the country could be put under Tier 4 measures - equivalent to full lockdown.
London and south-east England were placed into the Tier on Sunday - devastating festive plans for many - with no household mixing allowed unless within support bubbles.
Mr Jenrick told ITV News that ministers would review the situation on Tier restrictions but no changes to the framework would be made before 25 December.
"It's the day before Christmas Eve, people have made plans," he said.
"There will be further discussions today, led by the Prime Minister, as to whether the new variant requires us to take further steps," Mr Jenrick said.
"We need to make sure that the Tiered system, that was designed prior to learning about the new variant, is sufficiently robust, can do the job, and can ensure that people are protected all over the country."
Labour leader Sir Keir Starmer has written to the prime minister to say his party would back any government moves to tighten restrictions if that is what scientists recommended.
Jonathan Ashworth: 'We're in a race against time to save lives'
Speaking to ITV News on Wednesday, shadow health secretary Jonathan Ashworth urged Boris Johnson not to "dither and delay like last time" if advised to impose tougher rules.
"We're getting close to the April peak and we're saying to Boris Johnson: 'Don't dither and delay like last time. If you're advised to take tougher restrictions, get on with it.'"
Mr Ashworth continued: "We need to save lives, minimise harm, and then we need to roll out the vaccination programme rapidly.
"Let's put rocket boosters underneath that vaccination programme because we're in a race against time now to save lives and keep people safe."
ITV News Correspondent Ben Chapman reports from Stoke - an area that could be moved to Tier 4
The Daily Telegraph reported local leaders and health officials in Birmingham met on Tuesday night to discuss the possibility of the city being moved into Tier 4.
With warnings areas in lower tiers could be moved up to Tier 3.
The measures may come too little too late, however, with the new Covid variant - first identified in September - having already spread around the UK according to genomic researchers.
Experts believe the new strain is 70% more infectious than the one previously being dealt with.
The outbreak has caused countries around the world to close their borders to the UK, with France stopping freight travel earlier in the week too.
Health chiefs in Cumbria have said the new variant is already in the county and could be behind sharp increases in new cases.
Elsewhere Lancashire’s director of public health said there was a “high likelihood” the new variant was in the county.
Explained: What are the new Covid-19 restrictions and which areas are in Tier 4:
In Ireland, Sinn Fein has said an island-wide travel shutout should be introduced to stem the spread of the new variant there.
While existing travel bans between Scotland and England look set to be tightened from Boxing Day to try and stop the spread.