Nadhim Zahawi says supply is tight, but he expects the UK to meet its February target of 15 million doses.
Another 7,700 registered with coronavirus on the death certificate brings the total to nearly 104,000.
Schools Minister Nick Gibb says schools and families will have two weeks' notice for reopening.
Comedians Meera Syal, Romesh Ranganathan and Adil Ray make a video urging people to get the vaccine.
The NHS says any invitation which asks for vaccine payment or bank account details is a scam.