A spokesman said he was moved on the advice of his medical team and was receiving "excellent care".
Mr Johnson has asked Foreign Secretary Dominic Raab to deputise "where necessary", the spokesman added.
The prime minister, 55, was admitted to hospital in London with "persistent symptoms" on Sunday evening.
The Queen has been kept informed about Mr Johnson's health by No 10, according to Buckingham Palace.
BBC political correspondent Chris Mason said the prime minister was given oxygen late on Monday afternoon, before being taken to intensive care.

However, he has not been put on a ventilator.
A No 10 statement read: "The prime minister has been under the care of doctors at St Thomas' Hospital, in London, after being admitted with persistent symptoms of coronavirus.
"Over the course of [Monday] afternoon, the condition of the prime minister has worsened and, on the advice of his medical team, he has been moved to the intensive care unit at the hospital."
It continued: "The PM is receiving excellent care, and thanks all NHS staff for their hard work and dedication."
Mr Raab - who will later chair the daily Covid-19 meeting - said there was an "incredibly strong team spirit" behind the prime minister.
He added that he and his colleagues were making sure they implemented plans Mr Johnson had instructed them to deliver "as soon as possible".
US President Donald Trump said Americans "are all praying for his recovery".
He described Mr Johnson as "a very good friend of mine and a friend to our nation" who is "strong" and "doesn't give up".
Mr Johnson was initially taken to hospital for routine tests after testing positive for coronavirus 10 days ago. His symptoms included a high temperature and a cough.
Earlier on Monday, he tweeted that he was in "good spirits".

We've been told he is still conscious, but his condition has worsened over the course of the afternoon.
And he has been moved to intensive care as a precaution in case he needs ventilation to get through this illness.
Last month, the prime minister's spokesman said if Mr Johnson was unwell and unable to work, Mr Raab, as the first secretary of state, would stand in.

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