The elder Bush served as the US president from 1989 to 1993.
Bush served as Vice President to Ronald Reagan in the 1980s, and in 1988 won the Republican nomination for president.
Trump tweeted his statement from Buenos Aires, where he arrived on Thursday for the G20 summit.
"Our family has had a front-row seat to the most awesome love story", Jeb Bush said.
His pronouncement soon proved prophetic, with the fall of the Berlin Wall and the collapse of the Soviet Union occurring early in his presidency.
His wife Barbara Bush died in April.
The statement concludes: "Our hearts ache with his loss, and we, with the American people, send our prayers to the entire Bush family, as we honor the life and legacy of 41".
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"George H W Bush was a man of the highest character and the best dad a son or daughter could ask for".
Only one other USA president, John Adams, had a son who also became president.
Mr Bush had vascular parkinsonism, a form of Parkinson's disease, and used a wheelchair or motorised scooter for the last six years.
Thousands of USA troops were sent into Panama and the country's leader, Manuel Noriega, was arrested on drug trafficking charges and extradited to the U.S. to face trial.
He assembled a coalition of 32 nations to drive Iraqi forces out in a matter of weeks with a lightning air and ground assault. He was succeeded by Democrat Bill Clinton.
Bush, the 41st president of the United States, lived longer than any of his predecessors.
His father, Prescott Bush, the son of an OH steel magnate, made his fortune as an investment banker and later served 10 years as a senator from CT. "But I see history as a book with many pages, and each day we fill a page with acts of hopefulness and meaning".