Mr Johnson, 55, has been given oxygen but hasn’t been placed on a ventilator, cabinet minister Michael Gove said.
Foreign Secretary Dominic Raab has been asked to deputize for the PM. Arriving at No 10 on Tuesday, he said the prime minister was “in very good hands”.
World leaders have sent messages to Mr Johnson wishing him well.
The prime minister, 55, was admitted to St Thomas’ Hospital with”persistent symptoms” on Sunday evening and proceeded into the intensive care ward on Monday at 19:00 BST.
He was transferred as a precaution so that he could be near a ventilator – that takes over the body’s breathing procedure – BBC political correspondent Chris Mason said.
Mr Gove told BBC Radio 4’s Today programme on Tuesday:”The prime minister is not on a ventilator. He’s received oxygen ”
If there’s a change in his state”No 10 will ensure the nation is upgraded”, Mr Gove added.
The Queen was kept informed about Mr Johnson’s wellbeing, Buckingham Palace said.
As the first secretary of state, Mr Raab is the ministry designated to stand in for Mr Johnson when he’s unwell and not able to work.
Mr Raab arrived at No 10 on Tuesday morning and will afterwards chair of the government’s daily Covid-19 meeting.
He said before there was an”incredibly powerful team spirit” behind the prime minister and that he and his colleagues were making sure they implemented plans Mr Johnson had instructed them to deliver”when possible”.
Former Conservative leader Sir Iain Duncan Smith said the government will”continue to work” as decisions are made collectively by the cupboard.
He also sounded a warning to those who have broken social distancing guidelines, stating”if the most powerful man in Britain can come down with this, so can you”.
Mr Johnson was originally taken to hospital for tests after announcing 11 days ago he had the coronavirus. His symptoms included a higher fever and a cough.
Earlier on Monday, he tweeted that he was in”good spirits”.
Mr Johnson’s friend and former leadership of communications May Walden told BBC Radio 4’s Today programme Mr Johnson is”far fitter than he appears”.
“He will whip anyone’s buttocks on a tennis court, he runs frequently, he does not smoke, he drinks moderately.
“So I think if anybody is in a fantastic position both physically and emotionally to fight the disease then the prime minister is that individual.”
After very, very little information was shared now, the prime minister was taken into intensive care at about 19:00 BST.
We have been told he’s still conscious, but his condition has worsened over the course of the day.
And he’s been transferred to intensive care as a precaution in the event he needs ventilation to get through this illness.
The statement from Downing Street makes sure that he is getting excellent care and he wishes to thank each the NHS staff.
But something important has changed, and he’s felt it necessary to request his foreign secretary to deputise for him needs be.
That’s a very different message from what we have heard over the last 18 hours or so, where it was always”the prime minister is connected” and”he’s accountable” – almost like what is business as usual.
But obviously being in intensive care affects everything.
It comes as the amount of coronavirus-hospital deaths in britain reached 5,373 – an increase of 439 in a day.
The Department of Health and Social Care said that there were 51,608 confirmed coronavirus cases.
Also in hospital with coronavirus is veteran Labour MP for Rochdale Tony Lloyd, 70, who’s being treated at Manchester Royal Infirmary.
Intensive care is where physicians care for the sickest patients.
We don’t know the complete details of the prime minister’s condition, but his admission to ICU is the clearest sign of how sick he is.
Approximately two-thirds of patients admitted to intensive care of Covid-19 will require sedation and ventilation within 24 hours of arriving.
This is a disease that attacks the lungs and can lead to pneumonia and difficulty breathing. The body is left unable to get enough oxygen to the blood and into the body’s vital organs.
Boris Johnson has been given additional oxygen support.
There’s absolutely no proven drug treatment for Covid-19, even though there are lots of experimental candidates.
However, the cornerstone of the prime minister’s care will depend on getting enough oxygen into his body and encouraging his organs while his immune system fights the virus.
Among those who’ve sent messages to Mr Johnson was Labour leader Sir Keir Starmer, who described it as”horribly sad news”.
“All the nation’s thoughts are with the prime minister and his family during this incredibly difficult time,” he added.
Meanwhile, US President Donald Trump said Americans”are praying for his recovery”, describing Mr Johnson as”a very good friend of mine and a friend to our state” who are”powerful” and”don’t give up”.
French President Emmanuel Macron stated he sent”all of my service to Boris Johnson, to his loved ones as well as the British people in this difficult moment”.
Chancellor Rishi Sunak said his thoughts were with the prime minister and his pregnant spouse, Carrie Symonds, and that Mr Johnson would”come out of the even stronger”.
On Saturday, Ms Symonds said she’d spent a week in bed with the primary symptoms. She said she hadn’t been tested for the virus.
Scotland’s First Minister Nicola Sturgeon stated she was”sending [Mr Johnson] every good wish”, while Northern Ireland’s First Minister Arlene Foster added she had been”praying for a complete and speedy recovery”.
The Taoiseach – Irish Prime Minister – Leo Varadkar wanted Mr Johnson”a quick return to health”, while European Commission President Ursula von der Leyen also desired him a”speedy and complete recovery”.
For Archbishop of Canterbury Justin Welby, the information”deepens our empathy for all who are seriously sick” and people looking after them.
And Mayor of London Sadiq Khan tweeted that St Thomas’ Hospital had”some of the best medical staff in the world” and the prime minister”could not be in safer hands”.