Ministers are facing pressure to decide whether tourists returning to England from Greece should quarantine, after Scotland and Wales introduced their own measures.
Scotland has said travellers arriving from Greece must quarantine for 14 days after 04:00 BST on Thursday.
And Wales has asked arrivals from the Greek island Zakynthos to quarantine.
It comes as travel company Tui cancelled all holidays to a resort on the Greek island after virus clusters.
The firm said it would no longer be offering trips to the resort of Laganas, on Zakynthos, from Thursday because of customers failing to follow coronavirus safety measures.
Six clusters of cases have been linked to flights from the island - also known as Zante.
Greece has insisted it is doing "everything in our power" to keep UK holidaymakers safe.
The country's tourism minister, Haris Theoharis, told BBC Radio 4's Today programme: "We're taking specific targeted measures where we see specific concentrations of cases. Measures that have been successful and have been working in the past few days."
Wales' Health Minister Vaughan Gething said there were six clusters of cases, amounting to more than 30 infections, linked to flights from the Greek island in the past week - including two flights which landed in England.
He called for an "early meeting with the UK government and devolved nations" to discuss changing the rules.
His comments came shortly after the Scottish government announced its own decision.
Justice Secretary Humza Yousaf said: "With Scotland's relatively low infection rate, importation of new cases from Greece is a significant risk to public health."
Struggling to keep up? You're not alone! The UK government's travel quarantine has been a messy affair.
There was real chaos when restrictions were re-imposed on arrivals from Spain at the height of a busy summer weekend.
After that, ministers tried to bring in a more structured approach by reviewing each country's infection rate throughout each week and announcing any changes on Thursdays or Fridays.
However the picture has been complicated further by the fact that the quarantine is a public health policy and so the Welsh and Scottish Governments can diverge from Westminster and classify countries differently.
Now the Welsh government is bringing in testing on arrival for passengers too.
That's awkward for the UK government because, for months, the aviation sector has been asking ministers to give their backing to testing at airports so people who test negative wouldn't have to quarantine for the full 14 days.
Aviation bosses are frustrated about the delay over any announcement on that.
And all the twists and turns over quarantine makes life for any foreign travel business incredibly hard.
Meanwhile, the UK's biggest tour operator Tui is suspending its holidays in the resort of Laganas on Zakynthos.
Andrew Flintham, managing director of Tui UK and Ireland, said anyone who had booked to go the resort after 3 September would be refunded for the cancellation, but that trips to all other resorts on Zakynthos would continue as normal.
He said: "The health and safety of our colleagues and customers is our primary concern and recent cases show that some customers are not following social distancing and Covid safety measures.
"It is therefore the right thing to do to protect and reduce a now identified potential risk to others by no longer offering holidays to this specific resort."
Nearly 200 people faced self-isolation after at least 16 passengers on a Tui flight from Zakynthos to Cardiff Airport tested positive for the virus. Some people claimed passengers were not following Covid-19 rules.
When a country surpasses 20 cases per 100,000 people in the past week, the UK government normally imposes 14 days of self-isolation on returning travellers.
There were 14.3 coronavirus cases per 100,000 people in Greece in the seven days to 1 September, up from 14.1 a week earlier. But several cases of the virus in Scotland have been traced back to Greece, including a passenger on a Tui flight from Zante to Glasgow on 23 August.
Tui said customers due to travel to Greece from Scotland would be able to amend or cancel their holiday in light of the quarantine announcement.
Meanwhile, ministers are considering reimposing quarantine measures for those arriving in the UK from Portugal as coronavirus cases rise, sources have told the BBC.
It has been less than two weeks since a travel corridor was established between Portugal and the UK, following a sustained period of falling cases in the country that put it below 20 cases per 100,000 people.
But in the seven days up to 1 September, the seven-day rate has increased from 14.2 to 22.7.
Every year, more than two million Britons visit Portugal, making up the largest number of overseas tourists to the country.
Over May and June, the Portuguese government reopened its restaurants, coffee shops, museums and beaches. Hotels have mainly reopened, but nightclubs remain closed.
The government has not commented on whether requirements for arrivals from Portugal will change again.
Last week, Switzerland, Jamaica and the Czech Republic joined France, Spain and other destinations on the UK's quarantine list.