Mr Johnson, 55, is “in very good hands”, said Foreign Secretary Dominic Raab, who is standing in for the PM and chaired the government’s daily meeting.
World leaders have sent messages to Mr Johnson wishing him well.
It comes as Cabinet Office minister Michael Gove became the latest politician to go into quarantine.
Mr Gove said he was self-isolating at home, after a family member showed mild symptoms. He is not suffering any symptoms and will work from home.
He was moved as a precaution so he could be close to a ventilator – which takes over the body’s breathing process, NobleReporters learnt
Mr Gove said: “The prime minister’s not on a ventilator. He has received oxygen support.”
If there is any change in his condition “No 10 will ensure the country is updated”, Mr Gove added.
As the first secretary of state, Mr Raab is the minister designated to stand in for Mr Johnson if he is unwell and unable to work.
Mr Raab said earlier there was an “incredibly strong 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 “as soon as possible”.
Mr Johnson is the first major world leader to have announced he had the virus
Some politicians have called for greater clarity on what Mr Raab’s role as deputy entails, including Tory MP Tobias Ellwood who asked for details “as to where responsibility for UK national security decisions now lies”.
Lord Heseltine, who served as deputy prime minister under John Major, said it will be a “very difficult personal position” for Mr Raab, who “will be tested by the loneliness of the job”.
“He will be surrounded by lots of people who know what Boris Johnson said, believe Boris will be quickly back and have their own personal agendas anyway,” he said.
Former Conservative leader Sir Iain Duncan Smith said the government will “continue to work” as decisions are made collectively by the cabinet.
He also sounded a warning to people who have broken social distancing guidelines, saying: “I hope people who may have wandered out the other day and decided they can sit around having barbecues will really think about this carefully and recognise this is serious.
“If the most powerful man in Britain can come down with this, so can you”.
Mr Johnson was initially taken to hospital for tests after announcing 11 days ago that he had the coronavirus. His symptoms included a high temperature and a cough.
Earlier on Monday, he tweeted he was in “good spirits”.
Mr Gove is the latest cabinet minister to self-isolate, after Mr Johnson, Health Secretary Matt Hancock and Scottish secretary Alister Jack.
The government’s chief medical adviser Prof Chris Whitty and the PM’s adviser Dominic Cummings also spent time self-isolating after showing symptoms.
The Queen has been kept informed about Mr Johnson’s health, Buckingham Palace said. She also issued a message thanking healthcare workers for their “selfless commitment and diligence” to mark World Health Day.
It comes as the number of coronavirus hospital deaths in the UK reached 5,373 – an increase of 439 in a day.
The Department of Health and Social Care said there were now 51,608 confirmed coronavirus cases.
Separate figures from the Office for National Statistics have been released, showing the majority of coronavirus deaths are happening in hospitals but some are in hospices and care homes.
Among those who have sent messages to Mr Johnson was Labour leader Sir Keir Starmer, who 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.
Meanwhile, Russia’s Vladimir Putin wished Mr Johnson a speedy recovery, saying he was “convinced that your energy, optimism and sense of humour will help you overcome the disease”.
US President Donald Trump said Americans “are all praying for his recovery”, describing Mr Johnson as “a very good friend of mine and a friend to our nation” who is “strong” and “doesn’t give up”.
Chancellor Rishi Sunak said his thoughts were with the prime minister and his pregnant partner, Carrie Symonds, and that Mr Johnson would “come out of this even stronger”.
On Saturday, Ms Symonds said she had spent a week in bed with the main symptoms. She said she had not been tested for the virus.
Scotland’s First Minister Nicola Sturgeon said she was “sending [Mr Johnson] every good wish”, while Northern Ireland’s First Minister Arlene Foster added she was “praying for a full and speedy recovery”.