Holidaymakers are being urged not to jet off to amber list countries, including Spain, despite the country lifting travel restrictions.
Business minister Anne-Marie Trevelyan said scientists still think there is "too great a risk" in travelling to amber list countries for non-urgent reasons such as holidays.
Spain has officially lifted restrictions for UK travellers from Monday, with visitors no longer needing to take a PCR Covid-19 test.
However, tourists going to Spain will still have to quarantine on their return under UK rules.
Mrs Trevelyan told Times Radio: "The reality is, at the moment, amber countries are still not meeting the criteria for our scientists to say that they should be green.
"So the recommendation remains don't go unless you have to, and remember that, if you do go, you will have to quarantine for 10 days and that will be monitored.
"The reason we ask people still not to go is because there is still too great a risk as far as our scientists are concerned."
Mrs Trevelyan said the Prime Minister "has been clear" that people should not travel to amber countries unless there is a pressing reason, such as an urgent family need.
England is "still trying to slowly move through our road map to being able to open up on June 21 and we want to do that in a steady and careful way," she told Sky News.
Mrs Trevelyan said it is hoped that "the amber numbers will become more green in due course" but "at the moment, today, that means amber countries really aren't safe to go to".
When the green list was confirmed it was announced the countries on the list would be reviewed every three weeks.
This would mean the next update will be on June 7.
It came as an expert from the University of Oxford said the pandemic could be declared over if vaccines manage to keep people out of hospital.
Professor Andrew Pollard, director of the Oxford Vaccine Group, was discussing Public Health England (PHE) data published at the weekend showing that two doses of Pfizer or AstraZeneca are "highly effective" against the Indian variant.
He told BBC Radio 4's Today programme that more time is needed to see how the vaccines work in the longer term as people build immunity.
Asked if people may think the Oxford/AstraZeneca vaccine is not as good as Pfizer, according to the data, he said: "Well, I think in some ways we're looking at the wrong exam question, because this is a question about mild infection and transmission.
"But the thing that makes this a pandemic is people going into hospital. And so what we really need to know, and we don't have the data yet for certain, is how well both vaccines are performing in preventing people from going into hospital.
"And what we've seen so far in the pandemic is that protection from vaccines against hospitalisation and death is much, much higher than the protection against mild infection, which is what these (PHE) tests are detecting.
"So what I'm waiting for is the answer to that exam question, which is the critical one to understand how we should respond in the future to new variants, is to find out whether these infections that we're starting to see a little bit with this current variant are completely uncoupled from hospitalisations and deaths, but we just need a few more weeks to get more evidence around that."
Asked if the pandemic could be declared over if hospital admissions can be kept low, Prof Pollard said: "If the current generation of vaccines are able to stop people going into hospital, whilst there is still mild infections, people are getting the common cold with the virus, then the pandemic is over.
"Because we can live with the virus; in fact, we are going to have to live with the virus in one way or another, but it doesn't matter if most people are kept out of hospital because then the NHS can continue to function and life will be back to normal. We just need a little bit more time to have certainty around this."
Asked whether booster vaccines will be needed, he said work is continuing but it is not certain that they will be.
He urged people to have their second jab, adding that the virus will find the unvaccinated and will adapt to become more transmissible.
"Of course this is the variant that's around at the moment but future variants are going to get even better at doing that," he said.
"That's the evolution of this virus, that it's going to find ways around immune responses to be able to spread a bit better, and so that gives a really important public health message, which is that if you're unvaccinated, then the virus will eventually find those individuals in the population who are unvaccinated, and of course if you're over 50 and unvaccinated, you're at much greater risk of severe disease."
