According to the Smoking Gun website, the hacker, who went by the online name "Guccifer," got access to the family's private information, including emails, photos, private telephone numbers and addresses of Bush family members and friends.
The website posted photos it said came from the hacker, including one that purported to show the elder Bush during his recent stay in a Houston hospital, where the 88-year-old spent months being treated for a bronchial infection.
The authenticity of the photos and other details on the website were not immediately confirmed. A spokesman for former President George H.W. Bush declined to comment on the reports.
The Secret Service said the agency was investigating and would not elaborate.
The Associated Press contributed to this report.