Authorities in France, Germany, the U.K. and Italy are looking to avoid another blanket closure of schools this autumn, relying instead on steps such as social distancing and mask-wearing to contain infections. In case of outbreaks, they plan to shut down only individual classes or schools.

The stance generally has support from unions, as well as many parents, and is bolstered by the absence of school-related outbreaks in day-care centers and elementary schools that remained open last spring, when infection levels were far higher.

In recent weeks, daily new cases have risen in countries including Germany, France and Spain. But while Europe as a whole is now reporting about 12,000 cases a day—more than 2½ times as many as in early July—that is well below the 32,000 a day recorded at the peak in April. It also is far lower than the 53,000-a-day seven-day average recently in the U.S.

In the German state of Mecklenburg-Western Pomerania, where the school year started last week, two schools temporarily shut down after a teacher at the first and a pupil at the second were found to be infected, underlining the challenges ahead.

But for now, authorities are undeterred. Classes have been divided into clusters, with students allowed to interact with each other but not outside the group. One such group was quarantined at a school in the city of Rostock after several members of a family tested positive, but the school remained open.

“Nothing has changed. On the contrary, our precautionary concept is working, and we are focusing on targeted measures to prevent renewed blanket closures,” said Henning Lipski, spokesman for the Mecklenburg Western-Pomerania government.

Kay Czerwinski, head of the parents association in Mecklenburg-Western Pomerania, said schools should remain open.

“We have to persevere. Children—especially in elementary schools—must return to in-person teaching as soon as possible,” he said. “Everything else is untenable.”

Teaching, Mr. Czerwinski said, is based on the interaction between students and teachers. He cited experts who say remaining at home is impeding children’s mental development. And many parents can’t go to work if their children are at home, he added.

In the U.S., calls by President Trump to reopen schools have been met with opposition from some experts and media amid an intense debate about whether such a move would boost contagion, especially given the significantly higher rates of disease incidence across much of the nation.

In Europe, pressure is high to return children to the classroom so that parents can go back to work. Policy makers are also concerned about the impact of prolonged home schooling on students, especially in poorer families.

“School closures are only effective if we want to damage our children,” said Wieland Kiess, a professor of pediatrics at the Leipzig Research Center for Early Child Development in Germany. He coordinated a study that showed isolation at home is damaging the mental health of children, especially those from poorer families.

In Germany, back-to-school rules vary from state to state. Children in Mecklenburg-Western Pomerania must wear masks on school buses and all common areas outside of classrooms. Classes aren’t allowed to mix on school premises. Teachers are encouraged to take complementary coronavirus tests.

North Rhine-Wesphalia, where the summer break ended this week, imposed a masking order during class for all high-school students. In Berlin, which also reopened Monday, children must wear masks when moving around the building but not in classrooms.

In Scotland, students returning this week are being kept in groups throughout the day to limit intermingling of different age groups and expected to regularly wash their hands. Face coverings aren’t compulsory, but older children and adults may be asked to wear them if data point to an increase in infections in the surrounding community.

Some scientists have warned against broad reopening of schools, pointing to school outbreaks outside Europe. Israel has recorded several clusters, mainly in high schools. In the U.S., hundreds of students age 6 to 19 became infected at a summer camp in Georgia in June.

Many disease experts say the risk to children from Covid-19, the disease caused by the coronavirus, is small, with multiple studies showing most display only mild symptoms, if any. Studies also have indicated that younger children haven’t been driving the epidemic.

Peter Klimek, assistant professor at the Section for Science of Complex Systems at the Medical University of Vienna, coordinated a study of pandemic measures in 76 countries and territories that found school closures to be one of the most efficient measures in curbing contagion among the community at large.

However, Prof. Klimek said this effect could be the result of other factors, such as parents having to work from home while taking care of their children. That means the parents have fewer outside contacts and thus fewer opportunities to become infected themselves.