J.P. Morgan Chase & Co. and Wells Fargo & Co. are snarling the flow of data to popular websites that help consumers manage their finances, according to people familiar with the matter.

Banks, facing increasing competition from these companies, are becoming more protective of their customer information and are limiting how much data they pass on. The moves also reflect growing concern within the banking industry that rising use of such sites will overload bank servers, on top of worries that customer data could potentially...