Health chiefs are under growing pressure to extend jabs to the over-12s before schools go back in September.
Around 86% of parents with children under 16 would 'definitely or probably' say yes to them having the Covid vaccine, according to the Office of National Statistics.
Ministers and school leaders are understood to be keen to help avoid further disruption to schooling.
The Joint Committee on Vaccination and Immunisation announced last week that 16 to 17-year olds would get their first dose to help curb infection rates.
NHS England said nearly 16,000 people older teenagers have already received their vaccine over the weekend.
The JCVI is waiting for more data on safety and effectiveness before deciding if younger secondary age children should also be offered a first jab.
Many countries have already decided to vaccinate children over the age of 12 - including Canada, Brazil, the US, France, the Netherlands and Italy.
Danny Altmann, professor of immunology at Imperial College London, said vaccinating all teenagers would be "a good thing" as the more unvaccinated people there were, the more lungs there were "for virus to percolate in".
The jabs expert said children who have the virus but do not have symptoms are "as dangerous to the spread as anybody else".
Prof Altmann added: "From a medical scientific point of view, I'd say there's nothing special about the virus in their lungs that can't transmit through to their families, through to their schoolteachers, through to their colleagues".
His comments came after Professor Sir Andrew Pollard said pupils who are not unwell should not have to isolate after being exposed to a positive case.
Sir Andrew, who helped to create the Oxford/AstraZeneca vaccine and is a professor of paediatric infection at the University of Oxford, said that as long as children who are contacts are not ill, they should continue to be in school getting their education.
Some children have already been jabbed because of their own health conditions or because they live with vulnerable adults.
