Depository Trust & Clearing Corp., a firm at the center of Wall Street’s trading infrastructure, is about to give the technology behind bitcoin a big test: seeing whether it can be used to bolster the $2.6 trillion repo market.

DTCC said in a statement Tuesday that it will begin testing an application of blockchain, the digital ledger originally used to track ownership and payments of the cryptocurrency bitcoin, to help smooth over problems in the crucial but increasingly illiquid corner of short-term lending markets known...