U.S. law enforcement agencies are probing Denmark’s largest bank over allegations of massive money laundering flows from Russia and former Soviet states, according to a person familiar with the matter and documents reviewed by The Wall Street Journal.

The Justice Department, Treasury Department and Securities and Exchange Commission are each examining Danske Bank after a confidential whistleblower complaint was filed to the SEC more than two years ago, the person familiar said. The probes are ongoing and related to transactions...