Four pupils at a primary school which Prince George and Princess Charlotte attend are in self-isolation after a trip to northern Italy, it is understood.
The four from Thomas's Battersea school in southwest London have had tests and are staying at home as a precautionary measure.
They are now awaiting the test results. None have been diagnosed with COVID-19.
It comes as at least eight schools in the UK have shut and others have sent some pupils home over fears they may have been exposed to COVID-19 - the disease caused by the coronavirus - during ski trips to northern Italy.
A spokesman for Thomas's Battersea said: "Like all schools we are taking the potential risks connected with the spread of COVID-19 very seriously.
"And to this end are following government guidance to the letter around both prevention against infection and in dealing with cases where any staff or pupils are suspected of being exposed to the virus or who display any symptoms.
"We currently have a very small number of pupils who have been tested and these individuals are currently, as per government advice, remaining at home pending the receipt of their test results.
"All parents have been informed and we have maintained regular communication with our school community to ensure that advice is shared and important information is circulated.
"We will of course preserve staff and pupil confidentiality and will not be commenting on specific cases."
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.
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.
St Peter's C of E middle school in Old Windsor, Berkshire, said on social media it had closed for the day as well.
Cransley School in Northwich, Cheshire, and Trinity Catholic College in Middlesbrough announced on Tuesday they would be closed for the rest of the week to allow a deep clean.
Lutton St Nicholas and Gedney Church End primary schools in Lincolnshire also said they had closed "because of a potential connection to the coronavirus by an individual within the school".
St Christopher's C of E High School in Accrington told parents it would be shut on Wednesday.
Meanwhile, Sandbach High School in Cheshire said students and staff who visited Aprica, in Italy's Lombardy region, were told to stay indoors and self-isolate.
A third Cheshire school, Brine Leas School in Nantwich, said its sixth form was closed due to staff shortages following government advice regarding travel to Italy.
Students from Penair School in Truro, Cornwall, Salendine Nook High School in Huddersfield, Cardinal Heenan Catholic High School in West Derby and Newquay Tretherras in Newquay, have also been advised to stay home after returning from ski trips.