Around 70 per cent of parents said they were worried about the risk, with only three in ten untroubled.
Worry increases slightly among mothers with children at school, 74 per cent of whom are worried compared to 64 per cent of fathers.
Those with children at secondary school are more likely to worry, with 72 per cent concerned about the risk of their children catching Covid while 68 per cent of primary school parents feel the same.
Children in Scotland returned to the classroom in the middle of August, while coronavirus cases are currently at record levels. English, Welsh and Northern Irish schools return early this month.
The Joint Committee for Vaccination and Immunisation (JCVI) recommended that 12 to 15-year-olds are only given the vaccination if they have underlying health conditions.
However, governments are able to make their own decisions based on information from their chief medical officer (CMO).
Scotland’s CMO is expected to update ministers later this week.
First Minister Nicola Sturgeon said last week that she has not ruled out giving Covid vaccinations in schools as she declared the health service was ready to move quickly if experts give the go ahead to jab 12 to 15-year-olds.
More than seven in ten people surveyed believe it is likely the number of coronavirus cases will rise as a result of children returning to school, while only 8 per cent deem this unlikely.
Kelly Beaver, managing director of public affairs at Ipsos MORI, which carried out the survey, said: “As pupils return to schools across the country, it’s clear that there is a certain degree of concern and trepidation about the impact of that on Covid cases, as well as concern among parents about the more direct impacts on them and their children.
"There are also broader educational concerns about the way that exams will be run next year and about the degree to which pupils will have ‘caught up’ on time lost academically from last year. These are important issues for parents, who’ll be watching closely to see what the government does around this crucial area.”
While almost two-thirds surveyed think most school staff will take lateral-flow tests twice a week, as the government currently advises, only half of secondary school pupils are likely to so the same.
Less than half of Britons expect to see exams run as normal in 2022, but 22 per cent think this is unlikely. Parents are slightly more hopeful, with 51 per cent saying exams are likely to run as usual.
More than four in ten believe the government will once again be forced to close schools because of Covid, while one in four disagree.
Almost half say it is unlikely that by the end of the coming academic year, pupils will have caught up with any education missed since the start of the pandemic. Only a quarter think this is likely to happen, increasing to a third of parents.