Education Secretary Gavin Williamson is pushing for the jab to be rolled out to children between 12 and 15 as pupils return to school across the region following the summer holidays.
Heads fear Covid outbreaks could lead to children missing out on yet more of their education following the chaos of the last 18 months.
Currently the jab is only being offered to over-16s but the Joint Committee on Vaccination and Immunisation (JCVI) is now coming under pressure to extend it to younger children, as has already happened in the US and parts of Europe.
Andrew Clewer, headteacher at Landywood School in Great Wyrley, said: “They missed so much school during the last academic year and I think it’s time for children to have that level of protection to try and ensure they get an academic year without disruption.
"And if children test positive that’s almost unavoidable. You are still going to have individuals who come into contact and if they haven’t got the vaccine they will end up missing even more school. We are trying to get children back on track.”
Mr Clewer said he did not believe there would be resistance from parents to younger children getting the vaccine.
Mr Williamson. who is also the MP for South Staffordshire, said offering the vaccine to children would be a "crucial step" in ensuring the new school term is not blighted by Covid outbreaks.
He told the Express & Star that the move will help to ensure schools can keep classrooms full when pupils return next week, and that parents needed to have a choice over offering the jab to their children.
Mr Williams said: "We obviously have to wait for the JVCI, but I think it would be incredibly reassuring for parents to know that they have the choice about whether to get their children vaccinated.
"As a parent myself, I see it as a really crucial step for the return of schools and it is guided by the best scientific and medical advice.
"The vaccination programme here is one of the most successful in the world. We are seeing other countries vaccinating children from age 12 and we are starting to see the benefit in that.
"We are ready to start moving forward on this as soon as we get the go ahead."
It comes after Scotland's deputy first minister, John Swinney, said a surge in cases north of the border was partly being fuelled by the return of schools after the summer holidays.
Public Health England's medical director Dr Yvonne Doyle insisted schools were not "drivers" of Covid.