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.”
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Mr Raab – who will later chair the government’s 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”.
“That’s the way we’ll bring the whole country through the coronavirus challenge,” he said.
Labour leader Sir Keir Starmer described it as “terribly sad news”.
“All the country’s thoughts are with the prime minister and his family during this incredibly difficult time,” he added.
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”.Skip Twitter post by @BorisJohnson
Boris Johnson #StayHomeSaveLives✔@BorisJohnson
Last night, on the advice of my doctor, I went into hospital for some routine tests as I’m still experiencing coronavirus symptoms. I’m in good spirits and keeping in touch with my team, as we work together to fight this virus and keep everyone safe.135K7:20 AM – Apr 6, 2020Twitter Ads info and privacy32.7K people are talking about this