According to the Sun newspaper report published on Sunday, the 38-year-old Duke of Cambridge had told an observer at an engagement that "there were important things going on and I didn't want to worry anyone".
The Duke was treated by palace doctors, and followed government guidelines by isolating at the family home Anmer Hall, in Norfolk, the report said.
"William was hit pretty hard by the virus - it really knocked him for six," a source told the Sun newspaper.
"At one stage he was struggling to breathe, so obviously everyone around him was pretty panicked.
"After seeing medics and testing positive - which was obviously quite a shock given how fit and healthy he is - William was determined it should be business as usual though.
"He was determined to fulfil his engagements," the source added.
The source further said that after the news Prince Charles and Johnson's diagnoses broke, William was concerned that informing the country about his own plight would send everyone into panic mode.
"The Queen delivered her 'We Will Meet Again' address, and he just didn't want to worry people. He felt there were more important things going on in the country.
"But as a result of his own experiences, he realises absolutely anyone can catch this awful disease - and knows how imperative it is that we all take this second lockdown seriously," the source told the Sun newspaper.
Prince Charles isolated for two weeks in March with mild symptoms, while Johnson spent three nights in ICU after being hospitalised on April 5.
The Kensington Palace was yet to officially comment on the development.
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