It comes as the Prime Minister has been urged to set out how he will reintroduce face-to-face teaching.
Councillor Joyce Underhill, Sandwell Council's cabinet member for Best Start in Life, said: "As with everything the Government keep changing the goal post with it and they don't tell us what they're going to change without sufficient time to put things in place which is very unsettling for teachers, children and parents who have to get children to school and arrange childcare.
"I think it's just so unsettled, first they said it would be half term, I think that was too optimistic, now they're talking about Easter but things have to be reassessed to see whether there's going to be a downturn in Covid because we don't want to think it's under control and get back too early and end up back to square one.
"It's very unsettling but it's doing the best we can and giving them as much notice as we can when children are ready to go to school.
"It's an unusual situation and we've got to try and make the best of it that we can.
"I know children would rather be at school with their friends but I am sure that parents have explained to them about Covid."
Councillor Chris Towe, Walsall Council's cabinet member for education and skills, said it was "disappointing" when he heard the Education Secretary Gavin Williamson had said he hopes schools in England can fully reopen to pupils before Easter.
He said: "I don't have access to the information the Government has, obviously I want children back in schools, it's the best place for them.
"They're doing brilliant work to give them their education.
"I do want them back in schools but I don't want them back in schools if there's going to be an issue.
"We haven't got teachers vaccinated yet, that is something we've asked for as clearly that would be helpful and make them feel more confident.
"It was disappointing when I heard that they hoped to be back before Easter as it is quite some time away.
"Education Secretary Gavin Williamson has told parents they'll get two weeks notice.
"Last time was a bit of a disaster, they went back for one day then were told they couldn't go back."
Councillor Ruth Buttery, cabinet member for children and young people at Dudley Council, is calling for teachers to be vaccinated.
She said: “We all know the vital role schools play in our children’s education and wellbeing. The government must therefore prioritise the full reopening of our schools and other education facilities as soon as it is safe to do so.
“Our schools are working hard to prepare for fully reopening when the government gives us the go ahead and in the meantime we are calling for the urgent vaccination of teachers and essential school staff to help protect them against Covid-19.”
Councillor Michael Hardacre, Wolverhampton Council's cabinet member for education and skills, said: "For the sake of children's mental, physical and emotional health it's far better that they are with their peers however in the current circumstances the Government is asking people to keep their children at home.
"All of us would want to see schools open as soon as possible but that has to be in conditions which Public Health, the Government and local authorities deem to be appropriately healthy and safe."