WASHINGTON—Firms offering online loans, smartphone payments and other financial-technology products would get new flexibility to expand and further shake up the U.S. banking industry under a proposed new federal policy.

A top regulator said Friday that his agency would for the first time start granting banking licenses to “fintech” firms, giving them greater freedom to operate across the country without seeking state-by-state permission or joining with brick-and-mortar banks.

The...