Mr. Cordray has been moving toward writing new rules on this front. But it is uncertain if he and the bureau will survive and push the banks to open up under the Trump administration.

Banks, in the meantime, have taken the initiative by pushing technology companies to accept new agreements on how they use the data they pull from the banks.

One of the primary companies that help move data between the banks and the start-ups is Envestnet Yodlee. The company said that in the last two months, several large banks had told it that it would lose access to at least some data in the near future if it did not agree to new restrictions on the data it is pulling.

Some of the banks have said they do not want to share the interest rates and fees that they charge customers, even when customers ask for that information to be passed along, said Steve Boms, the vice president for government affairs at Yodlee. Mr. Boms said that his company was pushing back against the requests. “With data limitations, you are hindering the ability of millions of consumers to save more and optimize their finances,” he said.

JPMorgan and Wells Fargo, which have been among the most aggressive in seeking new agreements, said they would pass along any information that customers wanted, as long as the customers themselves requested it. JPMorgan is hoping to create a dashboard on its website where people can choose to turn on or off the data flowing from the bank to any outside provider.

The banks say they are pushing for new data agreements in an effort to stop technology companies from getting access to customer data in ways that the customers might not understand, or that could create security risks.

Right now, few rules or standards exist for how technology companies can use the data they collect from customers. It is also not entirely clear who would be held liable if a data breach at a service like Venmo or Mint led to financial losses for a customer.