That describes a process known to private investigators and others as "pretexting" in which a person falsely uses personal information of another to obtain even more information about that person.
The Secret Service told CNBC on Tuesday that it was investigating the incident and it would provide no further information. A spokeswoman for the FBI, whose Director Robert Mueller was apparently one of the victims of the info leak, said only that it is aware of the website in question. NBC News and CNBC are not identifying the site that posted the government official and celebrity information.
Information apparently obtained from a TransUnion credit report for Mrs. Obama included her social security number, former addresses, former phone numbers, details of an account at Bank of America, a Chase Manhattan Mortgage and an apparent credit account at Banana Republic.
(Read More: US Sharing Classified Information to Stop Hack Attacks)
If the information is accurate, it is apparently stale. A call by CNBC to one of the phone numbers reached the voicemail for a person affiliated with the University of Chicago Medical School.
Current and former government officials impacted by the data breach apparently include Vice President Joe Biden, former Secretary of State Hillary Clinton, Attorney General Eric Holder, former Alaska Gov. Sarah Palin and former Vice President Al Gore.
Other individuals apparently hacked include Kim Kardashian, Ashton Kutcher, Jay-Z, Paris Hilton, Hulk Hogan, Donald Trump and Arnold Schwarzenegger.
(Read More: As Hacking Against U.S. Rises, Experts Try to Pin Down Motive)
In his statement O'Neal said TransUnion, which refers to itself as a global leader in credit information and information management services, "is taking steps to assist the individuals affected to help minimize any potential impact."
"We are conducting our own internal investigation and working closely with law enforcement," he said.
—By CNBC's Eamon Javers; Follow him on Twitter: