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.
Mr. Johnson was admitted to St Thomas’ Hospital with "persistent symptoms" on Sunday and moved to intensive care on Monday at 19:00 BST.
He was moved as a precaution so he could be close to a ventilator - which takes over the body’s breathing process - BBC political correspondent Chris Mason said.