In a statement, a family spokesman said he had contracted an infection that spread to his blood, but was "responding to treatments".
The 93-year-old was admitted to hospital in Houston on Sunday morning, the statement said.
He was president from 1989 to 1993 and his son, George W Bush, went on to serve in the same role.
Barbara Bush, the only woman to see both her husband and her son sworn in as US president, died last Tuesday and was laid to rest on Saturday.
The couple had been married for 73 years.