The Smoking Gun website said the hacker, who went by the online moniker "Guccifer," gained access to emails, photos, private telephone numbers and addresses of Bush family members and friends.
The website displayed photos it said came from the hacker, including one that purported to show the elder Bush during his recent near-two-month stay in a Houston hospital where the 88-year-old was treated for complications arising from a bronchial infection.
The authenticity of the photos and other details on the website could not immediately be confirmed. A spokesman for former President George H.W. Bush declined to comment on the reports.
"There's a criminal investigation and, as such, there's nothing else we can say," Jim McGrath told The Associated Press Friday.
The FBI in Houston, where Bush lives, was similarly tight-lipped.
"We do not confirm or deny the existence of any investigation," Houston FBI spokeswoman Shauna Dunlap said.
Freddy Ford, a spokesman for former President George W. Bush, who has a home in Dallas, also said Friday he "won't be commenting."
The FBI in Dallas referred calls to the Secret Service, which did not immediately return a message left Friday morning.
The word "Guccifer" is plastered across the photos published on the Smoking Gun website in translucent, neon blue print. The website quotes "Guccifer" as describing himself as a veteran hacker who has long been in the government's sights.