US media report that he is in intensive care. 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.
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.
Tributes paid at Barbara Bush's funeral
Obituary: Barbara Bush - former US First Lady and literacy campaigner
The couple had been married for 73 years.
After her death, Mr Bush said of her: "We have faith she is in heaven and we know that life will go on, as she would have it. So cross the Bushes off your worry list."
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