The prime minister said a huge network of 233 hospitals, 1,000 GP surgeries, 200 pharmacies and 50 mass vaccination centres is already working "exceptionally fast", but "at the moment the limit is on supply" of the vaccine.
"We will be going to 24/7 as soon as we can," he told MPs during Prime Minister's Questions in the Commons, adding Health Secretary Matt Hancock would be setting out further details "in due course".
His comments came minutes after his vaccines minister Nadhim Zahawi confirmed the government is considering a 24-hour vaccination programme to meet its promise to have the UK's four most vulnerable groups vaccinated by the middle of next month.
Mr Zahawi told MPs on the Science and Technology Committee ministers "will absolutely look" at the measure when asked about it, adding that he is confident the government will achieve its target.
He refused to say how many doses the UK is receiving from manufacturers on a weekly basis because he did not want to "show off", and said it was also a matter of "national security".
Labour leader Sir Keir Starmer said he understood that pilot 24-hour centres were not yet open to the public but there would be a "huge clamour".
The NHS needs to accelerate inoculations to vaccinate 15 million people in just five weeks.
AstraZeneca said manufacture is a biological process that can't be accelerated, but the company is confident of supplying tens of millions of doses in the first quarter of 2021.
"We've released just over 1.1 million doses, to date, and we are scaling up as we've said very rapidly. And this will happen imminently, to releasing two million doses a week, we're absolutely on track to do that," said Tom Keith-Roach, president of AstraZeneca UK.
"We're scaling up to two million a week imminently, and certainly we hope to be there on or before the middle of February," he told the cross-party parliamentary committee.
It takes 58-60 days to grow the cells, then there is a further 28 days needed to turn the raw product into a filled vial of vaccine.
Each batch needs to pass 60 quality tests, including confirmation that the vaccine contains the correct genetic sequence for the coronavirus spike protein and a validation of purity.
The vaccine maker's research chief also gave evidence to the committee, telling MPs his staff should get priority access to the jab to avoid COVID outbreaks hindering production.
"If you have an outbreak at one of the centres - which we've had actually - or in one of the groups in Oxford is working on new variants, or the people that are working on the regulatory files, everything stops," Sir Mene Pangalos said.
"This is a concern that I have and so again we're pushing to try and get our key workers that are working on the vaccine project immunised to try and prevent these outbreaks."
He added that current data shows that an eight to 12 week interval for the second dose of the Oxford vaccine is a "sweet spot for efficacy".
The second dose of coronavirus vaccines are now being given three months later than originally planned to ensure more people are given a first dose to help fight the UK's rising COVID-19 infection rate.
Earlier, England's deputy chief medical officer Professor Jonathan Van-Tam defended the move to prioritise first vaccinations, rather than keeping doses to deliver booster shots after three weeks.
He told LBC Radio: "We have all got older loved ones and if we want to protect as many as we can as quickly as possible, with a meaningful amount of protection, then the right strategy for us is to give the initial first dose and come back for the second when we have given more people the initial first dose," he said.
"If you have got two grandparents and you have got two vaccines, what do you do - do you give two doses to one and leave the other one with nothing?"
Meanwhile, journalist and TV presenter Baroness Dame Joan Bakewell is threatening the government with legal action over delays to the second dose of the Pfizer/BioNTech vaccine.
The Labour life peer said there are grounds to show the decision taken by ministers to delay the second dose by up to 12 weeks is unlawful and breaches the conditions of its authorisation - including patients' "legitimate expectations", with them consenting to a course of medical treatment on the understanding they would get a second dose after 21 days.