The Oxford-AstraZeneca Covid vaccine is being rolled out to hundreds of GP-run vaccination sites in England.
As part of the biggest vaccination programme in NHS history, the aim is to offer jabs to most care home residents by the end of January.
By mid-February, the target is to vaccinate 13 million people in the top four priority groups.
Mr Hancock said the jab was now being supplied to GP practices across the country as he visited the Bloomsbury Surgery in central London.
The health secretary said he was “delighted” care homes residents would begin receiving their first Oxford jabs this week.
“This will ensure the most vulnerable are protected and will save tens of thousands of lives,” he said.
Around half a million doses of both vaccines are ready to be used this week, with millions more in the pipeline in the coming weeks.
Next week, seven major vaccination hubs across England are set to begin operating, including the Excel Centre in London and Millennium Point in Birmingham.