During trial stages, the coronavirus vaccines were only tested on adults meaning some groups of people including children and pregnant women have not been invited to receive the vaccine.
The move is being discussed as authorities push for maximum immunity across the country.
Currently, only children at very high risk of severe infection are offered a jab.
But sources have revealed children aged under 18 could be given the jab as soon as August, months earlier than expected.
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Researchers have used 300 volunteers to assess whether the coronavirus vaccine will produce a strong immune response in children aged six to 17.
The vaccine trial began at Oxford University, as well as its partner sites in London, Southampton and Bristol.
The trial will see up to 240 children receive the vaccine, while others will receive a control meningitis jab.
This control vaccine will be used to prove safety in children, as it may produce similar reactions as the coronavirus jab, including a sore arm, headache or a high temperature.
Responding to reports that children could be vaccinated from August, Professor Finn told Good Morning Britain: “As far as I know there has been no decision made to immunise children starting in August, or indeed any decision been taken to immunise children at all at this point.
“But it’s certainly something that we might need to do.”
He added: “If it does turn out to be necessary to immunise children, I think it is more likely that we would prioritise teenagers over younger children, simply because the evidence we have at the moment is that transmission of the virus is more likely to occur from and between teenagers who are a little bit more like adults.
“I think what we need to learn before that (is) what proportion of the population we need to immunise in order to get effective herd immunity and to suppress circulation of the virus.
“In order to do that, we need to have a clear understanding of how efficiently the vaccines actually interrupt infection and transmission, and that evidence is still on its way at the moment.”
The country is already vaccinating 16 and 17-year-olds after determining it was safe, but the UK is so far only vaccinating those under 18 if they have an underlying illness which makes them a priority.
Professor Finn added: “During that time we will see what goes on with variants, with the circulation of the virus, and then we’ll be able to make a decision whether children need to be immunised – we clearly won’t want to do that unless it’s necessary.
“But if it is necessary we will by then know whether the vaccines are entirely safe and effective and we’re giving the right dose and so on so that we go forward with that later in the year.”
A spokesman for the Department of Health said: “While clinical trials are underway to test the efficacy and safety of Covid-19 vaccines in children and young adults, these trials have not concluded yet.
“We will be guided by the advice of our experts on these issues including the independent JCVI.”