Mr Johnson spent Monday night on a critical care ward at St Thomas' Hospital in London having been admitted to hospital "as a precaution" the night before.
Political leaders from the UK and worldwide have sent messages of support for the Prime Minister.
Labour leader Sir Keir Starmer led messages of support from all the major party leaders, saying "all the country's thoughts are with the Prime Minister and his family during this incredibly difficult time".
Speaking on Tuesday morning, the newly elected leader of the opposition said he had been in touch with the Foreign Secretary and reassured "the business of Government will continue".
Mr Karmer also extended thoughts to the Prime Minister's pregnant fiancee Carrie Symonds.
Labour's Jonathan Ashworth extended his thoughts to the Prime Minister on Tuesday morning.
The shadow health secretary said "we're rooting for him," adding:
"He's not just a politician, he's also a father - with his fiancee, they're expecting a child - this must just be horrific for his loved ones".
Elsewhere, Sir Ed Davey sent his support to the Prime Minister and his family. The acting co-leader of the Liberal Democrats wished Mr Johnson a "speedy recovery," adding:
"This shows that this awful disease, this cruel disease doesn't discriminate".
Scotland First Minister Nicola Sturgeon said she was sending the prime minister "every good wish", while Northern Ireland First Minister Arlene Foster said she was "praying for a full and speedy recovery".
The mayor of London, Sadiq Khan, added his wishes for a "swift" return to health.
St Thomas' hospital, where Mr Johnson is being treated, "has some of the finest medical staff in the world, and he couldn’t be in safer hands," he added.
Mr Johnson had been working from isolation until his worsening symptoms led him to be sent to hospital, initially as a precautionary measure.
A number of Conservative colleagues urged their "boss" to get back to full health, while also offering their support to his fiancee Carrie Symonds, who herself is overcoming the virus.
Chancellor Rishi Sunak posted: "My thoughts tonight are with @BorisJohnson and @carriesymonds. I know he'll be getting the best care possible and will come out of this even stronger."
International trade minister Conor Burns, who previously served as Mr Johnson's parliamentary private secretary, tweeted: "You are a fighter and we need you back."
Former cabinet colleague Amber Rudd said she "received a characteristically ebullient, light-hearted, positive response from Boris over weekend when I wished him a speedy recovery".
"I now want to multiply those good wishes 100 fold and send him hopes and support for a healthy recovery."
While business minister Nadhim Zahawi tweeted: "Thoughts & prayers for @BorisJohnson & @carriesymonds and their family. I have known Boris for 20 years he is a fighter and will beat this virus."
Former chancellor Sajid Javid echoed the sentiment, responding: "Boris is one of the strongest people I know. Thinking of him, Carrie and his family tonight. Get well soon, the country needs you."
Friends and political opponents of times past also offered their support, with former prime minister David Cameron tweeting: "Thinking of @BorisJohnson and his family tonight. Get well soon. You are in great hands and we all want you safe, well and back in @10DowningStreet."
While Mr Johnon's former Prime Minister’s Questions opponent Jeremy Corbyn posted: "My thoughts are with Boris Johnson and his family tonight. Thanks to the #NHS staff for their hard work and dedication."
French president Emmanuel Macron was the first major leader to send his wishes, tweeting: "All my support to Boris Johnson, his family and the British people at this difficult time. I wish him well."
Spanish Prime Minister Pedro Sanchez, who is also battling a major coronavirus crisis, tweeted his "solidarity and wishes" to his UK counterpart.
"These are difficult days for our countries, but from strength and unity, we will be able to win this battle. A hug to all the British people," he added.
While Italian President Guiseppe Conte tweeted: "My thoughts go to @BorisJohnson and all the British people. I wish you a fast recovery. You have the whole Italian Government's support and my personal one. We are all confident that your Country will get through this difficult time."
Well wishers from elsewhere across the world included Japan's Prime Minister Shinzo Abe who tweeted: "To my dear friend @BorisJohnson , my thoughts and prayers are with you and your family, as you fight for a swift recovery. The people of Japan stand with the British people at this difficult time."
In the United States - the country with the highest number of Covid-19 cases worldwide - President Donald Trump said Americans are praying for Mr Johnson's recovery.
He told reporters at the White House that the British leader has "been a really good friend," adding: "He's been really something very special, strong, resolute, doesn't quit, doesn't give up.
"When you get brought into intensive care, that gets very, very serious with this particular disease."
Elsewhere Australian Prime Minister Scott Morrison tweeted: "Get well soon @BorisJohnson. Thinking of you, your family and all our UK friends at this tough time."
European Commission President Ursula von der Leyen wished the Prime Minister "a speedy and full recovery" and European Council President Charles Michel said: "Our thoughts are with his family and the British people at this difficult time."
In Israel, Prime Minister Benhamin Netanyahu tweeted: "Prime Minister Benjamin Netanyahu and the People of Israel pray for the speedy and full recovery of our friend British Prime Minister".
Back in the UK, the Archbishop of Canterbury said the news Mr Johnson has been moved to intensive care "deepens our compassion for all who are seriously ill and for those caring for them."
"I invite all people of faith to join me in praying for Boris Johnson and his loved ones," Justin Welby wrote on Twitter.
While the Archbishop of York John Sentamu said a candle has been lit for the Prime Minister and "everyone afflicted with Covid-19".
JK Rowling and Gary Lineker were among the celebrities sending messages of support to the Prime Minister.
Harry Potter author Rowling said: "Terrible news about the PM. Wishing him a speedy recovery and thinking of his poor family.
"Yet more evidence that this horrible virus doesn't discriminate: we need to protect each other and slow the spread by staying home."
Match Of The Day host Lineker tweeted: "Crikey. Shocking news. Hope he pulls through."
While ITV's Good Morning Britain host Piers Morgan said it is an "incredibly serious situation" for Mr Johnson.
"Fight hard Boris - we're all rooting for you," he added.
Confirming Mr Johnson had been moved to a critical unit, a number 10 spokesman said: "Over the course of this 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.
"The PM has asked Foreign Secretary Dominic Raab, who is the First Secretary of State, to deputise for him where necessary.
"The PM is receiving excellent care, and thanks all NHS staff for their hard work and dedication."
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