Mr Johnson, who spent the night in the intensive care unit (ICU) at St Thomas' Hospital after his Covid-19 symptoms worsened, has not required a ventilator to help him breathe.
While in the ICU Mr Johnson was tested for pneumonia but later given the all clear, contrary to rumours on social media.
Meanwhile, the Queen sent a message to Mr Johnson's family and his pregnant partner Carrie Symonds, saying they were in her thoughts and that she wished him a full and speedy recovery, Buckingham Palace said.
The PM was also wished a "speedy recovery" by Prince William, Prince Charles and his wife Camilla.
Mr Johnson was originally admitted to St Thomas' on Sunday on the advice of his doctor after continuing to display symptoms of cough and high temperature ten days after testing positive for the virus.
The PM's spokesman said: "The Prime Minister has been stable overnight and remains in good spirits.
"He is receiving standard oxygen treatment and breathing without any other assistance.
"He has not required mechanical ventilation or non-invasive respiratory support."
The update came after former prime minister David Cameron said he was "praying" his friend Mr Johnson would get through his illness.
He was echoed by Mr Johnson's predecessor Theresa May, who said her "thoughts and prayers" are with the prime minister.
She added: "I wish him well and I want him to have a speedy and good recovery."
Cabinet minister Brandon Lewis told ITV News Mr Johnson is "fully conscious" and "still the prime minister".
He said: "It's important for people to be very aware of the fact he's still the prime minister, he's very much conscious but Dominic, yes, who is empowered as the first secretary of state, has been asked to stand in and deputise, by the prime minister."
The spokesman also gave an update on the running of government in Mr Johnson's absence, saying Dominic Raab, who is "deputising" has the power, along with the cabinet, to respond to an attack on the UK in Mr Johnson ’s absence.
Mr Raab will also be able to take military action without the consent of the Prime Minister, the spokesman added.
But he will not be asked to step in to brief the monarch on Boris Johnson's behalf.
The Prime Minister's official spokesman said: "Buckingham Palace and Her Majesty the Queen have been kept regularly informed about the Prime Minister's condition and that will continue.
"The Cabinet Secretary and the Prime Minister's principal private secretary will continue their contacts with the Royal Household on the Prime Minister's behalf."
An intensive care consultant explains what ICU is and how patients are treated in critical care:
Reacting to US President Donald Trump's offer of American drugs to help Boris Johnson overcome coronavirus, Downing Street said medical care for the Prime Minister was a decision for his NHS doctors.
"We are confident the Prime Minister is receiving the best possible care from the National Health Service," the spokesman said.
"Any treatment he receives is a matter for his doctors."
Asked whether Mr Raab would field any phone calls with the Queen, the spokesman added: "No, it has been agreed with the Royal Household that weekly audiences will not go ahead."
Should Mr Raab be forced to self-isolate or take ill, Chancellor Rishi Sunak would be next in line to take over, Downing Street added.
"In line with the order of precedence, the Chancellor would follow from the Foreign Secretary," Mr Johnson's spokesman said.
Downing Street said it would look into whether to publish the list for the order of precedence for other Cabinet ministers.
Asked about reports of Mr Raab seen coughing on Tuesday morning, the spokesman added: "The Foreign Secretary is fine."
Coronavirus: Everything you need to know