The government says it is on track to have offered an injection by Monday to everyone who is aged 70 and over, as well as those who are clinically vulnerable, frontline health and social care workers and older adults in care homes. "I'm optimistic but we have to be cautious", he told reporters during a visit to a vaccine manufacturing facility in Teesside.
Johnson will outline his easing of the lockdown plan on 22 February, and he is expected to announce the reopening of schools for 8 March.
"Then working forwards to getting non-essential retail open as well, and then in due course as and when we can prudently and cautiously of course, we want to be opening hospitality as well", he said.
Britain, which has recorded more than 120,000 deaths from Covid-19, was the first Western country to begin mass vaccinations in December, and more than 14 million Britons have since received their first dose.
Some newspapers reported on Saturday pubs and restaurants might be able to serve to customers outdoors from as early as April, and restrictions on social mixing easing by May, but Mr Johnson declined to be drawn on that timetable.
"I do think that in due time it will become something that we simply live with. Some people will be more vulnerable than others - that's inevitable".
Professor Steven Riley, a member of the SPI-M modelling group, told BBC Radio 4 Today: "If for some reason we were to choose to just pretend it (coronavirus) wasn't here anymore, then there is the potential to go back to a wave that is a similar size to the one that we are in now". "We don't want to show that it is an excellent but not ideal vaccine by having another large wave in the United Kingdom", he told the BBC Radio 4 Today programme.
Meanwhile, NHS Confederation chairman Lord Adebowale expressed concern that March 8 was still too soon for schools to return. We need to do so very safely. I think mid or late March is when we should be reassessing.
It comes as scientists warn against lifting lockdown restrictions too quickly, even if the vaccination programme remains on target.
Former cabinet minister David Davis said: "There will come a point where there will be a death rate from Covid but it is at a normal level and then we have to cope with that. Obviously we still try to prevent it but we accept it".