The U.S. Justice Department proposed that Deutsche Bank AG pay $14 billion to settle a set of high-profile mortgage-securities probes stemming from the financial crisis, according to people familiar with the matter, a number that would rank among the largest of what other banks have paid to resolve similar claims and is well above what investors have been expecting.

The figure is described by people close to the negotiations between Deutsche Bank and the government as preliminary, and they said it came up in discussions between...