The letter, signed by Leeds City Council leader Judith Blake, executive member for learning, skills and employment Jonathan Pryor, as well as the city's five Labour MPs, says the move would be "in the best interests of pupils and the country".
Addressed to Nadhim Zahawi, the minister responsible for vaccine deployment, the letter states: "We are writing to you to ask that you give due consideration to prioritising teachers and schools staff for the delivery of the Covid-19 vaccine.
"It is of course vital that NHS and care home workers, as well as other priority groups most vulnerable to the virus, receive the vaccine first. We are asking that teachers and school staff form an additional priority group following this."
It follows similar calls from the NASUWT teaching union and the NAHT school leaders’ union
The letter continued. "Parents across the country who have been working from home while also homeschooling their children will know how difficult this has been. We are sure you recognise the incredibly important role school and teachers play and the importance of schools reopening as soon as can safely happen.
"Teachers are currently running both virtual home learning as well as looking after our most vulnerable children and key worker children in schools, a situation which is both unsustainable as well as meaning the same level of education is simply impossible.
"We would be very grateful if you could please give serious consideration to this request as we do believe it is in the best interests of pupils and the country."
The calls for teacher vaccination have been echoed across the profession.
Kevin Courtney, joint general secretary of the National Education Union, said: “We believe that education staff should be included in the vaccination programme at the same time as NHS and social care staff.
“This will both keep them safe when teaching in schools and colleges and make sure that education is less disrupted when schools can be safely re-opened beyond key worker and vulnerable children again.”
However, Mark Woolhouse, professor of infectious disease epidemiology, University of Edinburgh, told the PA news agency there is no data to suggest teachers are at greater risk.
Speaking as an independent researcher, and not in an advisory role, he said: “In terms of whether teachers are at risk, there’s been a number of studies on this now and teachers, as a profession, are not at more risk than many other professions, are certainly not at elevated risk either of getting infected in the first place or having a bad outcome of infection.”
He added that teachers who are at high risk will already be covered by the priority lists.
Prof Woolhouse said that, while children are infectious, most do not show symptoms and most experts consider children to be about a third as infectious as adults – depending on the age of the child.
He added: “When infections do occur in people in schools, whether they’re staff or students, that does not necessarily mean they got the infection in the school environment.
“So it’s perfectly possible for a teacher or school child to get infected outside the school.
“Every teacher that has turned positive in the UK, in recent months, not all of those teachers will have got infected in the school environment.
“If they do get infected in the school environment, the evidence is, certainly from Scotland, that they’re at least as likely to be infected from another teacher, as they are from a student.
“So you have to think about where the infection is actually being acquired.”
Prof Woolhouse told PA that, if vaccinating teachers was to become part of the strategy, then it would make sense to do so fairly quickly if the aim was to have them protected before the end of the February half term as it takes time for the vaccine to offer protection.
However, he added that this would mean potentially vaccinating teachers and staff over others on the priority list.
The Joint Committee on Vaccination and Immunisation (JCVI) currently puts older adults in a care home and care home workers, those aged 80 and over and health and social care workers, those 75 years of age and over, and all those who are 70 years of age and over and clinically extremely vulnerable individuals at the top of the priority list.
Professor Wei Shen Lim, Covid-19 chairman or JCVI, said: “The JCVI’s advice on Covid-19 vaccine prioritisation was developed with the aim of preventing as many deaths as possible.
“As the single greatest risk of death from Covid-19 is older age, prioritisation is primarily based on age.
“It is estimated that vaccinating everyone in the priority groups would prevent 99% of deaths, including those associated with occupational exposure to infection.”
Prof Woolhouse said that, while he did not know how the next phase of the vaccination programme was being worked out, it could take one of three approaches.
These could include continuing to go down the list by age, choosing the age groups that contribute most to transmission, or listing according to occupation.