Britain's prime minister also said at the press conference that restrictions on Britons travelling to see family would not be re-assessed until April.
The biggest threat to the UK and the one authorities are "all concerned about" in the country's fight against COVID are "variants of concern" being imported into the territory, England's Chief Medical Officer Chris Whitty said on Monday.
"These are COVID variants which might, and I want to stress the word 'might', have a problem with the vaccine where it is less effective against them," he added.
Whitty said that while "in the long term there would be ways around that" this was the concern that was "driving border issues".
British Prime Minister Boris Johnson said that travel restrictions - including on Britons wanting to visit family abroad - needed to remain in place to prevent new variants from entering the UK.
At the UK's regular COVID-19 briefing, Johnson was responding to a question from a member of the public who asked whether those who wanted to travel abroad to see family members could be treated the same way as travelling abroad for business, currently allowed under certain circumstances.
In response, the prime minister said that as the UK continued to vaccinate people against COVID-19 it needed to be sure that new variants did not get into the country from abroad.
"The most important thing that we've got to do now as we continue to immunise great numbers of people in this country is to protect our country insofar as we can [and] do as much as we can to prevent the virus coming back [and] new variants coming back from abroad," he said.
"So the rules about what you can do - what people can do to see their families - will be governed entirely by the rules that cover travel abroad and people coming from abroad, so at the moment, as you know, it is still forbidden for people to travel. We'll be saying a bit more in April."
Currently, 25 countries are on the UK's red list, meaning that people arriving from them need to be placed in a "quarantine hotel" for ten days when arriving in the UK.
Over 30 million Britons have received at least one dose of the COVID-19 vaccine, according to the UK Home Office, while 3.6 million have received two doses. But Johnson pointed to data that showed that while the vaccination drive prevented older people from dying, it did not prevent transmission.
That is because transmission is highest amongst younger people, a smaller proportion of which have been vaccinated.
Also speaking at the press conference Chris Whitty, England's Chief Medical Officer, urged all those who were vulnerable to COVID-19 to get the vaccine, arguing that it was their "obligation".