Mr Hancock said the next move down the age range will bring the country “a step closer” to seeing all adults offered a jab.
The Government has set a target for every adult in the UK to be offered a first dose by the end of July.
The news comes as people who have been contacted to bring forward their second coronavirus vaccine appointment are being urged to rebook as soon as possible.
The Joint Committee on Vaccination and Immunisation (JCVI) recommended in May that the second dose interval should be reduced from 12 weeks to eight for those aged 50 and over, as well as the clinically vulnerable.
The move aims to combat the spread of the Covid-19 variant first detected in India, also known as the Delta variant.
More than 40 million people across the UK have so far had a first dose of a vaccine.
A total of 57,510,753 Covid-19 vaccinations took place in England between December 8 and June 6, according to NHS England data, including first and second doses, which is a rise of 317,112 on the previous day.
NHS England said 33,800,107 were the first dose of a vaccine, a rise of 99,621 on the previous day, while 23,710,646 were a second dose, an increase of 217,491.
Mr Hancock did however tell MPs it was still too early to make decisions on step four of relaxing Covid rules.
A decision is expected to be announced next Monday, June 14.