School leaders have also said they have yet to be informed by the NHS whether children, aged 12 to 15, will definitely be offered the Pfizer-BioNTech vaccine following reports that its use for that age group has been approved by the regulator.
Health Secretary Matt Hancock said the UK has enough supplies to vaccinate children.
But the school bosses said accepting vaccination was a personal choice for all pupils and staff.
Chief executive of the Learning Community Trust, which includes three senior academies in Telford and Shrewsbury, Dr Gill Eatough said: "Parents can be reassured that we are not going to start vaccinating students without their consent.
"Vaccines are already administered in schools for other things, but like mass testing for coronavirus, administering jabs would be a huge undertaking when you have a school with 1,200 young people.
"I think it is early days for us to understand what the proposals might be. Like many things we have only hear about it first in the media. I have yet to hear from the NHS about what would be required.
"It is a difficult and complex issue in my opinion. It needs a lot of thought. Obviously parents will need to be sure that this is what they want for their young person."
Head of The Marches School, in Morda Road, Alison Pearson said: “Schools regularly host vaccination programmes for students, such as the HPV vaccination. Our role has always been to provide the venue for these only, with the NHS sending communication to parents and carers regarding consent and administering vaccinations.
"Schools have no part to play in these elements or making a judgment about the vaccinations being given. Vaccination is a personal choice for all staff and students.”
Medicines and Healthcare products Regulatory Agency chief executive Dr June Raine said the safety of the vaccine in 12 to 15-year-olds would be carefully monitored.
"No extension to an authorisation would be approved unless the expected standards of safety, quality and effectiveness have been met," she said.
Meanwhile, a Shropshire headteacher has welcomed suggestions that secondary schools children may be offered the coronavirus vaccine to combat infection rates.
School leaders in the county have yet to be informed by the NHS whether children, aged 12 to 15, will definitely be offered the Pfizer-BioNTech vaccine, but its use for that age group has been approved by the regulator.
Jules Taylor headteacher at Woodlands School, in Tilley Green, Wem, which is part of The Marches Academy Trust said if the move went ahead, parents would be encouraged to discuss it with their children first.
Mr Taylor said: “I think it is a good idea to vaccinate young people as currently rising rates seem to be amongst children in that age group and obviously this can then be spread to more vulnerable people who may, for whatever reason, not be fully vaccinated.
"Children themselves thankfully rarely seem to get very poorly with Covid-19 but vaccinating children will enable schools to be able to get back to normal as safely and quickly as we can.
"Parents and carers obviously have the right to decide, as with all vaccinations whether they want their child to be vaccinated.
"We would encourage parents to have a conversation with their child about how they feel about having the vaccine. We manage vaccination programmes annually for our students so this is not something that would be an issue if we are given the time to manage the logistics."
Liz Noakes, Telford and Wrekin's Director for Public Health, added: “The vaccination policy is decided nationally, based on evidence-based advice from the independent Joint Committee on Vaccination and Immunisation (JCVI).
"Our council will continue to support the vaccination programme locally by encouraging and helping people who meet the national criteria to access their vaccination – as the best way to protect themselves, their loved ones and the community.”
Health Secretary Matt Hancock, who signalled he was in favour of vaccinating 12-15-year-olds to prevent school outbreaks, told pupils returning to classrooms this week after the half-term break that the "most important thing" they could do was get tested to ensure they are not asymptomatically carrying the virus.
It comes as Downing Street faces a "difficult" choice this week on whether to relax all Covid measures on June 21 in England, with a decision due in the next seven days.
Mr Hancock said ministers were "absolutely open" to delaying unlocking of the Indian variant starts to fuel hospital admissions.
He also refused to rule-out social distancing and guidance on wearing face coverings staying in place beyond the target date.
Coronavirus cases have continued to increase in the UK amid reports the Prime Minister is considering pushing so-called freedom day back by at least two weeks in order to allow more people to be fully vaccinated against the renamed Delta strain.