The Good Morning Britain presenter and the MP agreed to take the approved vaccine together, if they were able to.
Hancock criticised what he views as a British cultural norm of going into work 'as long as you can get out of bed'.
A trial of a vaccine from Oxford University and AstraZeneca found it stops 70% of people developing COVID symptoms.
Robert Jenrick has said there is no 'expectation' to make the Covid vaccine mandatory, despite Matt Hancock refusing to rule it out.
Health secretary Matt Hancock was grilled by Piers Morgan in Good Morning Britain's first interview with a government minister in 201 days.
The government's boycott of 'Good Morning Britain' ended on Monday, and Piers Morgan took the opportunity to confront the health secretary over COVID.
Matt Hancock said Rashford and the prime minister two had been in contact, but the footballer said they have not spoken since June.
A coronavirus vaccine could be rolled out before 2021, the health secretary said, and a report claimed health staff could be jabbed before the new year.
Vast swathes of England have had their coronavirus tier-status raised as tighter lockdown restrictions loom for millions.
An Excel spreadsheet reaching its maximum size caused 15,841 infections between 25 September and 2 October to go unreported.
The government has admitted its coronavirus testing issues will take 'weeks' to solve.
Matt Hancock was jeered by MPs as he outlined Project Moonshot which would see millions of people tested for coronavirus everyday.
Matt Hancock said authorities have identified a number of pubs where the virus had spread, and now venues will need to close between 10pm and 5am.
Boris Johnson and Matt Hancock have both said the government didn't know COVID-19 was being transmitted by people with no symptoms. So, what's the truth?