A company spokesperson said: "Any customers that live in tier four and are due to depart in the next 14 days will be entitled to cancel and receive a full refund or amend for free to any holiday that's currently on sale."
The company said that cancellations do not apply to flights landing at Gatwick and Stansted, which are located in tier 2 areas.
Meanwhile, Easyjet (EZJ.L), also said it would fly its current schedule over the coming days.
"Following the UK government's announcement implementing tier four restrictions which includes advice against travelling abroad, we understand some customers may now need to change their flights," the airline told the BBC.
Virgin Atlantic also issued a statement on its Twitter account saying: “Customers who need to change their travel plans have the option to rebook, with a name change and two date change fees waived, for a new travel date up until 31 December 2022,” although the latest restrictions had not been added to their website at the time of writing.
British Airways and Virgin Atlantic are refusing to issue refunds to passengers from Tier 4 areas, unless their flights have been cancelled.
The news comes as multiple countries have moved to close their borders to travellers from the UK, as governments mitigate against a new mutant strain of the coronavirus.
On Saturday, UK prime minister Boris Johnson has announced stricter COVID-19 measures amid concerns about the spread of a new strain of the virus which may be up to 70% more transmissible.
In a press conference on Saturday, with England’s chief medical officer Chris Whitty and chief scientific adviser Sir Patrick Vallance, he said: “Residents must stay at home, with limited exemptions. People must work from home where they can.”
COVID-19 rules were due to be relaxed across the nation between 23 and 27 December, with up to three households able to meet across the five days.
So far, countries including Italy, the Netherlands, Austria, Belgium and Israel have announced that they will ban flights from the UK due to fears over the new variant of the coronavirus.
Watch: Should I pay off debt or save money during the COVID-19 pandemic?