There are hopes that every adult in Britain might be vaccinated by the end of June after jab delivery accelerates to five million a week.
Senior government figures are reportedly growing more optimistic about the vaccine rollout and believe everyone over-18 may get the crucial jab in five months.
So far, the only target set has been to give the jab to all over-70s, care home workers and NHS staff by mid-February.
The two million a week target appears to be on track after more than 324,000 doses were administered in the last 24 hours.
More than 3.5 million people in the UK have now received their first dose of a vaccine, Boris Johnson said, as he hailed those helping the “fantastic national effort”.
The Prime Minister tweeted: “We’ve given over 3.5 million vaccine doses to protect against COVID-19, with over 324,000 doses yesterday (Friday) alone.
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“Thank you to everyone who is helping in this fantastic national effort. Help our NHS by staying at home to save lives.”
But the Sunday Telegraph reports that Government figures are even more upbeat about the vaccine rollout behind closed doors.
They believe the rollout can be upscaled to between four and five million doses a week within months.
Two more vaccines, from Moderna and Johnson & Johnson, could speed this up.
A source told the paper: “All over-18s by June – yes. It is delivery, delivery, delivery.”
The Department of Health has said it hoped to have vaccinated “tens of millions” of Britons by April, but has not publicly acknowledged a deadline for every UK adult.
Government sources have said reports of a June deadline to vaccinate every British adult are speculative, the PA news agency reports.
Meanwhile, ministers are urging the public to “play their part” in supporting the vaccination programme, such as by helping the elderly attend their appointments.
Health Secretary Matt Hancock said people should also sign up to clinical trials for vaccines and treatments, and stay informed with accurate and trusted NHS advice.
He urged the public to commit to “three pledges” to support the rollout, saying: “Everyone has a part to play in this national effort – to protect our NHS, our loved ones and other people’s loved ones too.”
His plea came as another 1,295 deaths in the UK were reported on Saturday, the third-highest daily total since the pandemic began, but the lowest number of lab-confirmed cases this year was reported – 41,346.
The figures will be seen as a sign that infections may be levelling off or falling as a result of the lockdown measures, though scientists believe the peak in deaths will come later.
Also, this morning, the Sunday Times reports that a “Cabinet deal” has been done to approve a three-point plan for lifting coronavirus restrictions from early March.
The paper also believes ministers had been considering new border measures – including the creation of quarantine hotels and using technology to ensure people isolate.
NHS England has confirmed today that ten further mass vaccination centres will open in England this week with more than a million over-80s invited to receive their coronavirus jab.
Blackburn Cathedral, St Helens rugby ground, Norwich Food Court and a park-and-ride outside York are among the new locations where large-scale vaccination will take place from Monday.
These will join the seven existing mass vaccination sites across the country, alongside a thousand GP-led surgeries and more than 250 hospitals already providing jabs.