Both the new coronavirus and the seasonal flu virus are likely to spread in the fall and winter, and send many more sick patients to doctors’ offices and hospitals already struggling to treat Covid-19 cases, health and industry officials say.

To reduce the pressure, drugmakers including AstraZeneca AZN -0.11% PLC, GlaxoSmithKline GSK -0.62% PLC and Sanofi SA SNY -0.91% are making about 200 million flu shots this year for shipment to doctors, hospitals and pharmacies, up 13% from last year and a record, according to the U.S. Centers for Disease Control and Prevention.

The federal government is also launching a campaign encouraging people to get the shots, while drugmakers and pharmacies explore novel measures to ensure more people get vaccinated, like offering flu shots curbside.

“We don’t want there to be an overwhelming of the health-care system,” said Dr. Leonard Friedland, director of scientific affairs and public health for Glaxo, which last month began shipping about 50 million flu doses, up 10% from a year ago.