Video report by ITV News Health Correspondent Emily Morgan
Pupils at Prince George and Princess Charlotte's school are among those to have self-isolated after returning from coronavirus-hit northern Italy.
Four pupils from St Thomas's Battersea School in south-west London were told to stay home amid concerns they may have the Covid-19 virus.
It comes as several schools across the UK have closed, following the outbreak of the virus in Italy.
At least eight have closed, while others have sent pupils home amid fears they may have been exposed to coronavirus during trips to northern Italy.
However Public Health England (PHE) said that its general advice is not to close schools.
Italy has sought international support to help deal with the spread of the disease, as the number of cases in the country reached 400.
Italian premier Giuseppe Conte’s government appealed to European neighbours for cooperation, not isolation and discrimination.
The country has more coronavirus cases outside Asia than anywhere else.
“Viruses don’t know borders and they don’t stop at them,” Italian health minister Roberto Speranza said at the start of a crisis meeting with World Health Organisation and European Union representatives in Rome.
Twelve people infected with the virus have died in Italy since Friday, all of them elderly, having other health conditions or both, civil protection chief Angelo Borelli said.
The Italian government has defended its handling of the crisis, even as it acknowledges alarm over its increasing cases and inability to locate the origin of the two clusters in the northern Lombardy and Veneto regions.
The ContinU Plus Academy in Kidderminster said it was closed for the day on Wednesday after a staff member had been in "close contact" with a family member who was self-isolating following a trip to northern Italy.
A statement on its website said: "We are now taking advice from the DFE (Department for Education) and the NHS and hopefully will be able to reopen tomorrow.
"If a deep clean is deemed necessary we may be forced to extend the period of closure."
William Martin CofE Junior, Infant and Nursery School in Harlow, Essex, also said it had closed on Wednesday after a staff member had returned from Italy.
Headteacher Gina Bailey said: "The school is closed for one day and it is purely as a precaution.
"The staff member is not displaying any symptoms of the virus."
St Peter's C of E middle school in Old Windsor, Berkshire, said on social media it had closed for the day as well.
Its headteacher Andy Snipp said on Facebook: "I received information late last night that one of our students had returned from a half term break to a location which is broadly part of an affected area.
"I must repeat that this is simply a precaution and were I to have been aware sooner then we would have completed the clean prior to the opening of the school."
Pupils at Tudor Grange Academy, Kingshurst in Birmingham were sent home on Wednesday and the building closed "with immediate effect for a deep clean".
While the Lime Academy Watergall, in Peterborough has closed for the rest of the week.
The Six Nations match between Ireland and Italy was postponed as Europe struggles to maintain the outbreak.
The sides had been scheduled to play in Dublin on March 7 but the Irish Rugby Football Union (IRFU) announced the fixture had been postponed.
Ireland were also due to play Italy in an Under-20 Six Nations game and women's international that weekend.