New York prosecutors have demanded records relating to up to $16 million in loans that a bank run by a former campaign adviser for President Donald Trump made to former campaign chairman Paul Manafort, according to a person familiar with the matter.

The subpoena by the Manhattan district attorney’s office to the Federal Savings Bank, a small Chicago bank run by Steve Calk, sought information on loans the bank issued in November and January to Mr. Manafort and his wife, the person said. The loans were secured by two properties...