Pressure is mounting for England to consider imposing quarantine rules on travellers arriving from Greece amid reports of people returning with coroanvirus.
But Greece has insisted it is doing "everything in our power" to keep UK holidaymakers safe.
Buckinghamshire Council confirmed there were two groups of people who had tested positive for Covid-19 after returning from abroad.
One group from the Marlow area is understood to have been on a trip to the Greek island of Zante.
Meanwhile, a passenger on a flight back from Heraklion in Crete has said she was asked to self-isolate after it was believed someone on the plane had tested positive for the virus.
The woman, who wished to remain anonymous, said she had received a text from NHS Test and Trace saying she was in contact with someone on August 27.
But she told the PA news agency that the only place she was in contact with anyone where someone would have her contact details was the Tui flight back to London Stansted.
She added: "Our flight was hideous, hardly anyone wore a mask and the crew didn't do anything about it.
"We raised the issue twice with the cabin crew who still didn't enforce it.
"My family is OK but we are having to self-isolate.
"I'm so sad for my son because he has to miss his first week back at school."
A Tui spokeswoman said it was aware of a "small number" of positive Covid-19 test results following flight TOM5449 returning from Heraklion to Stansted on August 27.
She added: "We can confirm that no guests tested positive for Covid-19 in our hotels and no symptoms were displayed before or during the flight home by any of the customers."
In a statement, Buckinghamshire Council said a group of young people and some family members in the Marlow area had tested positive for the virus.
Cabinet member for education and skills Anita Cranmer said some of them were pupils at Sir William Borlase's Grammar School and all the affected households were self-isolating.
She added: "These cases are all linked and were acquired abroad during the school holidays.
"All households are now self-isolating having been contacted by test and trace staff.
"Public Health England are actively investigating these cases and have traced contacts to ensure that they are aware of the need to self-isolate."
It comes after around 60 students allegedly visited Zante in August, with several teenagers returning home with the illness, a parent told the Bucks Free Press.
The council also confirmed that another group of young people and their families, some of whom attend Chesham Grammar in Chesham, had also tested positive for Covid-19 and were self-isolating.
It added: "These cases are all linked and the majority of them were acquired abroad during the school holidays."
Rising coronavirus cases prompted the Scottish Government to announce travellers from Greece would have to self-isolate for 14 days from Thursday, while Wales began asking arrivals from the island of Zante to enter quarantine.
Public Health Wales has confirmed at least 16 cases of Covid-19 from three different parties who were on Tui flight 6215 from Zante to Cardiff on August 25.
One traveller claimed that the flight was full of "selfish 'covidiots'", with passengers not wearing their masks properly and disregarding the rules.
In a statement, Tui said that cabin crew and other passengers confirmed there were multiple warnings about wearing face masks on the flight.
Tui is suspending its holidays in the resort of Laganas on Zante.
There were 14.3 coronavirus cases per 100,000 people in Greece in the seven days to September 1, up from 14.1 a week earlier and a similar rate to the UK.
A seven-day rate of 20 is the threshold above which the UK Government has considered triggering quarantine conditions.
The Department for Transport (DfT) has said no change to its rules for England had been made, but Welsh health minister Vaughan Gething said he would be pressing the UK Government for an urgent meeting to consider the potential risk in Greece.
But Greece's tourism minister Harry Theoharis told BBC Radio 4's Today programme earlier on Wednesday: "We're doing everything in our power to ensure that every person that comes from the UK is kept safe in Greece.
"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."
Elsewhere, rising cases in Portugal are prompting many UK tourists to pay hundreds of pounds to fly home before Saturday.
In recent weeks the DfT has announced changes to its quarantine exemption list on Thursday nights, with the new rules being imposed on Saturdays at 4am.
In Portugal, the seven-day coronavirus cases rate is 22.7, compared with 14.2 for the previous week.
Holidaymakers hoping to avoid quarantine face limited options and expensive fares, despite airlines operating additional flights.
Figures have been calculated by the PA news agency based on data collected by the European Centre for Disease Prevention and Control.