Prime Minister Boris Johnson has stressed that any easing of restrictions needs to be in stages and in an "irreversible" way.
He will outline his "road map" out of lockdown on Monday, with schools expected to fully reopen on March 8,
But Chief Executive Officer of The Gorse Academies Trust, Sir John Townsley, said he was both '"concerned and surprised" that teachers remain "unprioritised" in the vaccination programme.
Sir John, who has a number of academies across Leeds, said: “It is clearly a national imperative for all pupils to return to school at the first opportunity.
"Nevertheless, such an undertaking requires a great deal of detailed and intelligent planning. Schools need to plan within guidelines provided well in advance by government and health specialists.
"We remain concerned and surprised that, in preparation for that return, teachers and other classroom professionals remain unprioritised in the vaccination programme.
"That programme will be central to our drive to ensure that all schools remain open in the months to come."
The council's executive member for learning skills and employment, Jonathan Pryor, said news earlier this month that the Oxford-AstraZeneca vaccine helps slow the spread of the virus only bolstered their case.
He added: "Just before the November lockdown it was just constant bubble collapses within schools. It was disruptive to parents. And actually, if we vaccinated teachers that disruption could be minimised massively.
"Sadly, they've chosen not to do that - but we have managed to vaccinate kind of any teachers that work with any clinically extremely vulnerable children."
He added: "If we make mistakes with the reopening of schools, it has the potential to reverse all the progress that everyone's made so many sacrifices for."
There have been reports that there will be tougher rules around mask wearing at schools and possible widespread testing within schools and at home to help contain the virus.
Leeds City Council has consistently urged all schools to encourage mask use everywhere possible.
On testing, Cllr Pryor said: "You never quite know kind of whether if people are doing them at home - are people going to be able do it properly, or possibly get a negative result when they're actually positive?
"It just feels like that putting the efforts in the wrong direction. Whereas if they focused on actually funding schools to make classrooms Covid secure and on vaccinating teachers, that would actually be effective.
"It would make so much more of a difference."
The Principal of The Grammar School at Leeds, Sue Woodroofe, told the YEP they "hugely miss" their pupils.
She added: "The past year has been a very difficult time for everyone and teaching children remotely is not the same as face to face.
"We should not forget that children learn so much from each other, not just from teachers, and social education is as powerful as academic learning.
"Schools have undoubtedly proven themselves to be very flexible and adaptable during the pandemic but we need time to plan and prepare so we have everything in place for the safe return of pupils.
"Schools, and indeed children, thrive on routine and structure, so when schools reopen we want them to remain open, to plan for a clear and positive trajectory out of this unprecedented situation, for the benefit of children, parents and teachers."
Matthew Partington, headteacher of Roundhay School, said it was wrong to set an "arbitrary date" for all children to return, and said instead that decisions "must be determined by the science".
He added: "We want every child to return to school as swiftly as possible.
"Concerns about increased transmission, impact on the NHS and progress with the vaccine rollout, must not be ignored.
"In the meantime, I know that all schools and families are working incredibly hard to ensure that pupils make progress whether in school or at home.
"I know from talking to other headteachers and principals, that we all agree that it is best to ignore the all too frequent rumours, and focus on supporting young people at this difficult time."