Twelve more patients in England have tested positive for coronavirus, taking the total number of UK cases to 35.
Three patients were confirmed as close contacts of a man from Surrey, who was the first to be infected within the UK.
One person from Essex had "no relevant travel" and it was unclear how they had got the virus, while the remaining eight had visited affected areas.
It comes after Health Secretary Matt Hancock said the UK was still in the "containment" phase of the outbreak.
He said the government contingency plans, which will be published this week, included banning big events.
On Friday, the man from Surrey, who had not been abroad recently, became the first person to be infected within the UK.
Three of the new cases announced on Sunday were linked to him – they included another person from Surrey and two people from West Sussex who were "part of an adult family cluster", Public Health England said.
Six of the other new cases had recently returned from Italy and two had been to Iran. They were from London, West Yorkshire, Greater Manchester, Hertfordshire and Gloucestershire and included:
- One person from Bury, who had become infected in Italy.
- Two Leeds residents, who had both been in Iran
- One person from Bradford, who had travelled to Italy
- Someone from Gloucestershire – infected in northern Italy – who is linked to another case from the area announced on Saturday
Prof Chris Whitty, the chief medical officer, said all of the newly identified cases were being investigated and health officials had begun tracing anyone who had close contact with them.
As of 09:00 GMT on Sunday, the Department of Health said a total of 11,750 people had been tested in the UK.
The latest positive tests come after three more cases were confirmed in England on Saturday.
One of those was a member of staff at St Mary's School in Tetbury, Gloucestershire, the primary school said on Sunday.
That follows another school – Willow Bank Infant School in Woodley, Berkshire – confirming that one of its staff had been infected.
The government has said no tactics will be "off the table" as part of its plan to contain the virus in the UK.
Mr Hancock told the BBC's Andrew Marr Show an emergency "battle plan", drawn up for the "worst case scenario", includes banning big events, closing schools and dissuading people from using public transport.
The health secretary said there were four phases to the plan:
- Containment – caring for any infected people and identifying their close contacts
- Delay – deciding what actions to take to slow down the spread
- Mitigation – damage limitation if the virus spreads widely
- Research – constant and ongoing work to inform the three other phases
Currently, the UK is in the "containment" phase – which health leaders say may still be sufficient.
But the next phase could see broader "social distancing" measures.
Matt Hancock would have been warned there was a strong likelihood there would be a jump in cases when he did his interview for the Andrew Marr Show.
A few hours later the biggest increase in a single day was announced, with 12 more people testing positive for the coronavirus. That's why he went further than before in setting out possible government plans to tackle an accelerated spread of the virus around the UK if that happens.
School closures, cancellations of major public events and moves to persuade people to work at home are clearly options being actively considered.
The strategy seems to be to warn that difficult measures may be needed while focusing on trying to contain the spread of the virus. As one government source put it, "prepare for the worst, work for the best".
Labour shadow health secretary Jonathan Ashworth said Mr Hancock must give more detail of any emergency powers he wants to bring in so that MPs "can properly scrutinise" the plans.
Meanwhile, Lib Dem MP Layla Moran has written to the health secretary to urge him to make sure people are paid if they have to self-isolate, otherwise there could be "devastating" effects.
Prime Minister Boris Johnson, who will chair a meeting of the government's Cobra emergency committee on Monday, visited Public Health England's centre in north London on Sunday to discuss attempts to limit the spread of the virus.
He said the virus was "likely to spread a bit more" but he was "very, very confident" the NHS would be able to cope with an outbreak.
More than 200 British holidaymakers locked down at a quarantined hotel in Tenerife have been told they will be allowed to fly home, if they test negative for the virus.
The travellers have been quarantined at the H10 Costa Adeje Palace hotel since Tuesday after four Italians contracted the virus. Some have already been allowed to leave but others are awaiting test results.
The Foreign Office is in contact with operators and travel agents about their plans to return Britons to the UK.