The King's cancer was "caught early", Rishi Sunak has said.
The prime minister said he was left "shocked and sad" when he was told about the King's cancer diagnosis.
"All our thoughts are with him and his family," he told BBC Radio 5 Live. "Thankfully, this has been caught early."
Mr Sunak said the running of the country will continue as normal and he was in "regular contact with the King".
The monarch, 75, will have to postpone or rearrange forthcoming public engagements following the diagnosis, according to the palace.
"His Majesty would like to apologise to all those who may be disappointed or inconvenienced as a consequence," a statement released on behalf of the King said.
The King remains "wholly positive" about his treatment for cancer and "looks forward to returning to full public duty as soon as possible", Buckingham Palace added.
The King has begun a schedule of regular treatments and is said to be receiving expert medical care from a specialist team.
He travelled from Sandringham in Norfolk to London on Monday to start treatment as an outpatient. He spent the evening at home in London, most likely in Clarence House.
The Queen will continue to attend to a full programme of public duties.
The King called both his sons, William and Harry, as well as his siblings the Princess Royal, the Duke of York and the Duke of Edinburgh, to give them the news before it was publicly announced.
The Duke of Sussex will be travelling to the UK from his home in the US in the coming days to visit his father.
However, it is understood that these duties will not relate to the King's constitutional role, and the monarch will continue to receive and review important documents in red boxes used by the monarch and government ministers.
It is understood that the King will continue to have weekly audiences with the prime minister while undergoing treatment, though these may be held remotely should medics advise him to reduce in-person contact.
He is also expected to remain available for meetings of the Privy Council, which usually meets once a month.
There has been no word yet, however, on upcoming overseas visits, with the King and Queen set to travel abroad for state events in the coming months.
Royal historian and author, Kate Williams, told Sky News the diagnosis would be "difficult" for the King.
"Charles is often one of the most hard-working royals with many engagements, many activities, many charities," she said.
"I think it is obviously going to be hard for him, because he won't be able to do this, and we don't know for how long."
It comes after it was announced the King had been diagnosed with a "form of cancer", which came to light while the monarch was in hospital for treatment for an enlarged prostate.
Though the palace did not elaborate on the type of cancer, they said it was not prostate cancer and described it as a "second condition".
A palace spokesperson said: "His Majesty has been treated for benign prostate enlargement.
"It was during this intervention that the separate issue of concern was noted and subsequently diagnosed as a form of cancer.
"This second condition will now receive appropriate treatment."
The King is said to have shared "his diagnosis to prevent speculation" and "in the hope it may assist public understanding for all those around the world who are affected by cancer".
News of the King's diagnosis comes after his daughter-in-law, the Princess of Wales, underwent abdominal surgery during a hospital stay of around two weeks.
Last month, Sarah, Duchess of York, the former wife of the King's brother, Prince Andrew, announced she had been diagnosed with skin cancer, just six months after being treated for breast cancer.