The Prime Minister will also hold talks with industry leaders in a bid to persuade them to start making desperately-needed ventilators to help hospitals treat those struck hown by the illness.
Health Secretary Matt Hancock was forced to deny that the Government wanted 60% of British people to contract the virus in order to ensure “herd immunity” against the outbreak.
Dramatic plans to isolate the country’s over-70s for four months were also leaked rather than being officially announced by ministers.
Labour leadership contenders Keir Starmer and Rebecca Long-Bailey, as well as Manchester mayor Andy Burnham, are among those who have demanded daily updates from Downing Street on the response to the outbreak, which has so far claimed 35 British deaths.
Sir Keir said: “To allow anonymous and speculative briefings to journalists about a significant step-change in the Government’s response to the outbreak is irresponsible.
“That is why I am urging the Prime Minister to promise an end to anonymous briefings about this issue.
“It’s time the Government commits to a daily press conference about its handling of coronavirus. This must be hosted by a Government minister or the Prime Minister himself, with experts and advisers available to answer questions as appropriate.”
From Monday, daily press conferences will be hosted by the Prime Minister and senior ministers, with scientific and medical experts including the Chief Medical Officer and Chief Scientific Adviser also taking part.
A Downing Street spokesperson said: “The Prime Minister and this Government are committed to keeping the public informed every step of the way about what we’re doing to fight the spread of coronavirus, when we’re doing it and why we’re doing it.
“At all times we will be led by the science to bring forward the right responses at the right time to this global pandemic.”
Mr Johnson will chair another meeting of the Government’s emergency Cobra committee on Monday, with discussions set to centre around isolating the elderly and vulnerable and banning mass gatherings of people.
On demands for industry to commit to producing ventilators for the NHS, a Number 10 spokesperson said: “Preparing for the spread of the coronavirus outbreak is a national priority and we’re calling on the manufacturing industry and all those with relevant expertise who might be able to help to come together to help the country tackle this national crisis.
“We need to step up production of vital equipment such as ventilators so that we can all help the most vulnerable, and we need businesses to come to us and help in this national effort.”