One photograph posted on the internet showed the 88-year-old Republican politician in bed in hospital, where he was recently treated for bronchitis.
The stolen emails are reported to include addresses and personal details of several members of the Bush family.
A spokesman for Mr Bush confirmed that an investigation was under way.
"We do not comment on matters under criminal investigation," Jim McGrath told the Houston Chronicle.
The hacker broke into email accounts of several members of the Bush family, news website the Smoking Gun reported.
The hacked emails are reported to include messages expressing serious concern about the health of the former president, including a personal note sent by President Barack Obama through an aide.
'Interesting mails'
Mr Bush was discharged from hospital on 14 January after a seven-week stay, during which he was treated for a bronchitis-related cough.
The purloined photos include pictures of his son, former President George W Bush.
One shows the younger Mr Bush posing beside a life-size cardboard cutout of himself with a moustache drawn on it; others are said to show paintings by him, including self-portraits.
The Smoking Gun said it had been in contact with the hacker, who goes by the alias Guccifer.
He said he taken "a lot of stuff" including "interesting mails" about the former president's time in hospital, the website reported.
George H W Bush was the 41st US President, serving one term from 1989-93. He was defeated by Democrat Bill Clinton when he stood for re-election.
Before becoming president he was Ronald Reagan's vice-president from 1981-89. He also served as vice-president, CIA director, US ambassador to China and congressman from Texas.
His son George W Bush served as 43rd president from 2001-09.
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