New York officials accuse Mr Ban's younger brother, Ban Ki-sang, and his nephew, Joo Hyun Bahn, of offering money to a Middle Eastern official.
They allege that bribes were given to entice the official to use government funding to buy their building project.
Ban Ki-moon served as UN secretary general from 2007 until 2016.
He was succeeded by former Portuguese Prime Minister Antonio Guterres on 1 January 2017.
Prosecutor say that in early 2013 the South Korean construction firm Keangnam, of which Mr Ban's brother Ban Ki-sang was an executive, was faced with growing debt and sought to sell a building complex in Vietnam known as Landmark 72.
In an 39-page indictment unsealed on Tuesday at a Manhattan courthouse, prosecutors alleged that he and his son Joo Hyan "Dennis" Bahn, a Manhattan estate agent, stood to make millions of dollars in commission on the sale, valued at up to $800m (Â£657m).
US officials say that the two men paid millions of dollars in bribes and tried to trade on the prominence of their South Korean family in order to persuade a Middle Eastern official from an unnamed country to arrange a purchase of the complex by that country's sovereign wealth fund.
Federal papers say the two even tried to secure a meeting to discuss the deal with the unnamed Middle Eastern country's head of state during his visit to New York during an annual UN general assembly meeting.
"This alleged bribery and fraud scheme offends all who believe in honest and transparent business," said Manhattan Attorney General Preet Bharara, "and it stands as a reminder that those who bring international corruption to New York City, as alleged here, will face the scrutiny of American law enforcement."
M Ban's relatives are charged with corruption, money laundering and conspiracy.
The brother and nephew of former United Nations Secretary General Ban Ki-moon have been charged with conspiracy to bribe a government official.