Coinbase Inc. and another cryptocurrency firm talked to U.S. regulators about the possibility of obtaining banking licenses, a move that would allow the startups to broaden the types of products they offer.

Coinbase, which operates the largest U.S. cryptocurrency exchange by trading volume, met with officials at the U.S. Office of the Comptroller of the Currency in early 2018, according to a person familiar with the matter. Meanwhile, Ivy Koin LLC, a payments startup, in recent weeks sat down with officials at the Federal...