PARK CITY, Utah—Financial-technology firms eager to offer banking products are eyeing a century-old model that fell out of favor during the financial crisis but could see a revival under the Trump administration.

The industrial loan company charter, available in a handful of states and particularly popular in Utah, allows nonfinancial companies to enter the banking sector without being subject to many of its restrictions, including oversight by the Federal Reserve. Companies seeking the charters must still obtain deposit insurance from the Federal Deposit Insurance Corp., which last approved insurance for an industrial loan company in 2008.