Most of the injections would be administered in the New Year, the Health Secretary cautioned.
But he said the recent spate of good news about the safety and effectiveness of a number of potential jabs had made him more optimistic for the future.
He said he had formally asked the regulator, the MHRA, to assess the Pfizer/BioNTech coronavirus vaccine for use in the UK.
The company has already begun submitting data to the regulator, he added.
The speed of any roll-out will depend on how quickly it is approved by the MHRA and how speedily it can be manufactured.
But he added: "If the regulator approves a vaccine we will be ready to start the vaccination next month with the bulk of roll out in the New Year.
"We are heading in the right direction but there is still a long way to go."
He added: "Chris Whitty (the chief medical officer for England) has said he hopes that we will be able to get back more towards normal by the spring.
"With this news we've had over the last few weeks and with the expansion of mass testing I'm more and more confident that he's right.
Earlier a leaked report suggested that vaccinations of all adults in England, no matter their age, could start before the end of January.
But professor Stephen Powis, the national medical director of NHS England, dismissed the document as out of date.
Vaccines will be administered based on age and need, such as underlying health conditions.
Prof Powis warned any roll out would take a number of months and that it was still important that the public complied with restrictions designed to fight the spread of the disease.
“We will have a tough winter ahead,” he warned.
Jonathan Van-Tam, England‘s deputy chief medical officer, tried to calm fears of those low down the priority list for the vaccine.
He said that if ministers were able to secure vaccines at the “ambition” they wanted there would only be a matter of weeks between the different levels on the government’s priority list.
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