They say the monarch is not showing any symptoms of coronavirus.
Prince Charles tested positive on Thursday morning - it is the second time he has contracted the virus.
The previous evening, he and the Duchess of Cornwall met Chancellor Rishi Sunak and others at a reception in the British Museum.
Camilla has tested negative for coronavirus, Clarence House said.
Buckingham Palace has not confirmed whether the Queen has tested positive or negative, citing medical privacy.
Earlier this week, Charles held an investiture at Windsor Castle, where the Queen, 95, is currently in residence.
The 73-year-old heir to the throne last caught the virus in March 2020, when he reported only mild symptoms.
Clarence House confirmed he was now triple vaccinated, but would not give any detail on whether he was experiencing coronavirus symptoms this time.
Tuesday's investiture carried out by the prince saw Ansel Wong, a Trinidadian-born cultural activist who previously helped to organise the Notting Hill carnival, receive a CBE for services to art and culture.
London restaurant founders and chefs Margot and Fergus Henderson were made OBEs for services to the culinary arts while Dr Nisreen Alwan, associate professor in public health at Southampton University, was made an MBE for services to medicine and public health during the pandemic.
As well as the investiture, it was a busy day for Charles with meetings and engagements including:
* A reception with Labour party leader Sir Keir Starmer
* A reception with Adm Sir Antony Radakin, upon assuming his appointment as Chief of the Defence Staff
* A dinner at St James's Palace in his capacity as Royal Founding Patron of the Professional Teaching Institute
On Wednesday, Charles and Camilla were at a British Museum reception to celebrate the work of the British Asian Trust.
Guests included the chancellor, Home Secretary Priti Patel and music producer Naughty Boy.
Earlier that day, Charles attended London's National Gallery as its royal patron.
People who develop coronavirus symptoms are asked to notify those they have been in close contact with in the previous 48 hours.
In England, contacts of positive Covid cases who are fully vaccinated should take lateral flow tests for seven days. They do not have to self-isolate unless they test positive.
Charles must self-isolate for 10 full days, but he could be allowed to end this early if he tests negative on day five and day six.
Prime Minister Boris Johnson has signalled that these rules could be lifted within weeks.
Charles had been due to unveil a statue of Licoricia of Winchester on Thursday, a medieval Jewish businesswoman who was a money lender and single parent in the Hampshire city.
Clarence House said His Royal Highness was "deeply disappointed not to be able to attend... and will look to reschedule his visit as soon as possible".
The announcement of Charles's positive test was made just after midday, about 12 minutes before he was due to arrive.
More than 1,000 people were gathered behind barriers in the city's Jewry Street, waiting to see the prince.
The Queen's representative in Hampshire, the Lord Lieutenant Nigel Atkinson, read out a message from the prince to the disappointed crowd.
Charles's message said: "Ladies and gentlemen, I am so sorry that I cannot be with you today. I am desperately disappointed as I was so looking forward to marking this historic occasion with you.
"I hope very much that I will be able to visit at a future time but for today please accept my most heartfelt apologies and my very best wishes as you mark this memorable occasion for Winchester."
Meanwhile, Camilla has carried on with her planned engagements, earlier visiting Paddington Haven, a sexual assault referral centre in West London.
She met former Love Island star Zara McDermott and discussed her experience of revenge porn and assault.
The Queen returned to Windsor Castle, where she met Charles, earlier this week after spending the anniversary of her accession day on Sunday at her Sandringham estate in Norfolk.
The day before, she had hosted a Platinum Jubilee reception for volunteer groups and pensioners from the Sandringham community, with guests describing her as being on "sparkling form".