Matt Hancock has said it is still 'too early' to say which restrictions could be eased in March following the first lockdown review.
The government is set to outline its plans for easing lockdown later this month.
Schools will be the first to reopen from March 8 at the earliest if the data allows, ministers have said.
Asked what else may be able to open in March, health secretary Mr Hancock told reporters today: "It is still too early to say - even though the vaccine programme is going great guns".
He said there were still "31,670 people in hospital with Covid as we speak right now" - which is "far, far too many".
“The NHS is still under pressure and we’ve all got to do our bit to keep those case rates coming down,” he added.
The government has said it is on track to reach its target of vaccinating everyone in the top four priority groups, as set out by the JVCI, by February 15.
Ministers will be looking at vaccine data and evidence around the impact of lockdown to inform decisions about how to ease restrictions.
Prime minister Boris Johnson has faced sustained pressure from some Tory backbenchers to relax the measures as soon as possible, but scientists advising the government have warned against opening up too quickly.
Professor Graham Medley, chairman of the Scientific Pandemic Influenza Group on Modelling (Spi-M), said ministers should “make decisions dependent on the circumstances, rather than being driven by a calendar of wanting to do things”.
But Mark Harper, chairman of the Covid Recovery Group, made up of lockdown-sceptic Conservative MPs, said it will be “almost impossible to justify having any restrictions in place at all” by the time the top nine groups have been vaccinated.
Meanwhile, Mr Hancock also confirmed plans for all those aged over 50 to have been offered a first vaccination dose by May.
“Lots of things have got to go right to hit that goal, especially supply, which is the rate-limiting factor," he said.
"But I’m sure, working with the NHS and everybody else who is making this happen, that if we keep going at the pace we can, then we can make sure all the over-50s get the offer of a vaccine by May.”
Asked about whether the UK will introduce 'vaccine passports' to facilitate international travel in the future, Mr Hancock said there were no current plans to do so.
Pressed on the matter by broadcasters, he said: “At the moment we have no plans for vaccine passports – and, anyway, going on holiday is illegal in this lockdown.
“Of course, we’re constantly working with other countries and we keep these sorts of things under review, but for now there are still 31,670 people in hospital with Covid right now.
“It’s imperative, that even with the good news on the vaccine and even with cases starting to fall, it’s imperative that everybody follows the rules and stays at home unless you absolutely have to leave.
“I know there is good news out there but we still have very serious problems that we’ve got to get through.”
