The Swift global banking cooperative countered Facebook and a legion of Bitcoin-inspired startups, saying it can continue to dominate the $73 trillion business of processing cross-border payments.

Disdaining blockchain, the software that underpins cryptocurrencies such as Bitcoin, Swift said in a report to be released June 20 that it has developed a system that is the “new standard” in payments.

A non-profit cooperative in Belgium of 11,000 institutions, Swift said a two-year-old service called Global Payments Initiative, or GPI, is now handling 55 percent of cross-border payments for banks including HSBC Holdings Plc and Citigroup Inc. Swift said all of...