The B.1.351 variant, also known as the Beta variant, has caused concern among health officials after a study suggested that the traditional Oxford/AstraZeneca jab was less effective when tested against it.
In a speech delivered at the Jenner Institute in Oxford, Mr Hancock said: "There is yet more to do, the work isn’t over yet - we’re still procuring all the time, and planning what we need to keep this country safe, including new vaccines specifically targeted at variants of concern."
"I can tell you today that we’ve started commercial negotiations with AstraZeneca to secure a variant vaccine – future supplies of the Oxford/AstraZeneca vaccine that have been adapted to tackle the B.1.351 variant first identified in South Africa.
"Once again, we’re leading the way and backing projects with potential, so we can keep our vaccination programme one step ahead of the virus and protect the progress that we’ve all made."
AstraZeneca has said that any future version of its vaccine would need to be approved for use by medicines regulators.
In a statement, the company said: "AstraZeneca is in discussions with the UK Government regarding future versions of our Covid-19 vaccine to specifically address variants.
"We will share more details in due course.
"Any future version of the vaccine would need to be approved for use by the medicines regulator."
Because of concerns about the variant, Test and Trace experts have jumped on any case found in the UK, and launched immediate rapid Covid-19 lab tests for people in areas where clusters have been identified.
Surge testing and so-called enhanced contact tracing has been deployed in a number of areas including parts of London.
To May 19, 904 cases have been identified in the UK – a rise of 41 from the previous week, according to Public Health England data.
Pfizer has previously said that there is currently no evidence that its jab needs updating against current known variants.
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