More than 14 million people in the UK have had at least one dose - with the aim being 15 million by Monday.
Meanwhile, Health Secretary Matt Hancock has said the UK could this year live with Covid "like we do with flu".
"I hope that Covid-19 will become a treatable disease by the end of the year," he told the Daily Telegraph.
Over the weekend, ministers are taking part in a renewed drive to encourage people in the top four priority groups - comprising people aged 70 and over, front-line health and social care workers, care home residents and the clinically extremely vulnerable - to get vaccinated.
The government is aiming to offer a vaccine to 15 million people in these priority groups by 15 February - which would cover 88% of those most at risk of dying from Covid-19.
On Friday, Wales said it was the first UK nation to meet the target.
Across the UK, a quarter of adults have already received one dose of a Covid vaccine, including around nine in 10 of all over 70s.
People aged 70 or over in England have already been asked to contact the NHS to arrange their jab. They can book an appointment online, or by calling 119 or contacting their local GP.
Health and social care workers should speak to their employer if they have not had their vaccine yet, the Department for Health said.
And GP teams have been asked to contact their clinically extremely vulnerable patients to make sure they have been offered a jab.
Almost 30 government ministers are taking part in visits and virtual meetings across the UK in a further push to encourage everyone eligible for the jab to get vaccinated.
They will hear from NHS staff and volunteers helping with the rollout, as well as people getting the vaccine.
Mr Hancock said: "I am determined that we protect as many of our country's most vulnerable people from this awful disease as soon as possible."
"Vaccines are the way out of this pandemic and it is testament to the strength of our Union and the combined power of our United Kingdom that we've seen such incredible progress in the roll out of our vaccination programme," he added.
Prof Stephen Powis, NHS medical director, said the UK's vaccination programme was "off to a strong start".
"People in the priority groups have not missed their chance to get jabbed and if you are aged 70 and over and haven't yet taken up the offer, please do come forward and make an appointment," he said.
The government has also published a new plan to help boost vaccine uptake in all communities, including those who may feel more hesitant about getting a jab.
The plan aims to raise awareness of how the NHS is making vaccination available to all, especially ethnic minorities, homeless people, asylum seekers and those with disabilities.
It highlights successful examples of community-led engagement which other areas could replicate, including a mobile vaccination service in Crawley to help reach those who are housebound and the use of social media resources to dispel vaccine myths in Slough.
From next week, details of who will be vaccinated in the next phase of the programme involving the over-50s and people with underlying health conditions will be set out. They are due to be vaccinated by May.
Some regions of the UK have already begun inviting the over-60s, who are part of the next phase, to be vaccinated.
On Friday, figures from the Office for National Statistics showed the number of Covid cases is going down in all nations of the UK - although infection levels remain high.
On Friday, 15,144 new cases were recorded in the UK, as well as 758 deaths within 28 days of a positive test.
Meanwhile, a new hotel quarantine system is due to come into force in England and Scotland on Monday.
It means all British and Irish citizens and UK residents arriving in England from so-called "red-list" countries - including Portugal, Brazil and South Africa - will have to quarantine in a hotel at a cost of £1,750 for an individual.
In Scotland, residents arriving from any country by air will have to isolate in hotels.
An eye for an eye will ultimately, leave the whole world blind.