NHS England said hundreds more GP-led and hospital services would also open later this week.
The government is aiming to vaccinate 15 million people in the UK – the over-70s, healthcare workers and those required to shield – by mid-February.
Health Secretary Matt Hancock will set out the government’s vaccine delivery plan at a press conference later.
He said the proposals would be the “keystone of our exit out of the pandemic”.
The government will also publish its first daily figures which will reveal how many people have been given the vaccine.
Mr Hancock said on Sunday about two million people in the UK had been vaccinated, with some 200,000 jabs administered in England daily.
The vaccine plan will be unveiled after the UK recorded more than 80,000 coronavirus deaths since the start of the pandemic.
In Surrey, which has one of the highest infection rates in the country, a temporary mortuary has been opened as hospital mortuaries have reached capacity.
Almost 200 bodies are being stored at the emergency site, which is a former military hospital, and other local authorities have told the BBC they expect to open similar facilities soon.
On Saturday scientists warned stricter lockdown measures might be needed in England and the health secretary has urged people to follow the spirit as well as the letter of the rules.
Mr Hancock told the BBC’s Andrew Marr Show on Sunday “every time you try to flex the rules that could be fatal” and said staying at home was the “most important thing we can do collectively as a society”.
Under the national lockdown, people in England must stay at home and can go out only for limited reasons such as food shopping, exercise, or work if they cannot do so from home. Similar measures are in place across much of Scotland, Wales and Northern Ireland.
Ministers held two meetings on Sunday to discuss how to enforce the current lockdown measures more strictly and whether even tighter restrictions may be needed.
BBC political correspondent Iain Watson said no decisions on further restrictions were taken as there was a desire within government to wait until reliable data on the impact of the existing measures becomes available in 10 days before going any further.
However, he added there had been a discussion on better enforcement of existing regulations which included how to ensure shops and workplaces that have remained open were observing health and safety rules, including social distancing. (Courtesy BBC)