LONDON—The world’s airlines are betting on vaccinations to restart international travel.

Two of Europe’s biggest airlines, British Airways and budget carrier Ryanair Holdings PLC, have started allowing fliers to provide vaccination and Covid test-result details alongside personal data, like passport numbers and visa information, during bookings. The airlines say the move will eventually help passengers show they have been inoculated when landing at destinations that have started to welcome vaccinated travelers.

Across the U.S., domestic travel is picking back up, amid stabilizing or falling Covid-19 cases and a relatively quick vaccination drive. That rebound isn’t yet showing up in international travel, where a patchwork of travel bans, quarantine rules and testing requirements have stymied cross-border flights.

U.S. domestic carriers have increased scheduled capacity by more than 50% between September and March, according to aviation analytics firm Cirium. Global capacity across all international routes, meanwhile, has increased just a little over 7%.

British Airways, Ryanair and other airlines dependent on international travel are hoping to boost ticket sales by capitalizing on nascent optimism over vaccinations. The move isn’t quite the sort of vaccination passport that some governments and international agencies are considering creating to help unlock pandemic-stricken economies. Countries have mulled documents that would allow vaccinated residents to visit bars and restaurants, or go to the office or a sporting event.