Political combatants set aside their fights to honour a Republican who led in a less toxic era and at times found commonality with Democrats despite sharp policy disagreements.
Another person told the Post that Bush "wouldn't want anyone there to feel uncomfortable, including the incumbent president'".
The late Mr Bush called Mr Trump a "blowhard".
The Oak Ridge Boys will perform "Amazing Grace" and Reba McEntire will sing "The Lord's Prayer" at Thursday's services at Houston's St. Martin's Episcopal Church.
During his 4 years in the White House, Bush ended the Iraqi occupation of Kuwait, steered the United States through the end of the Cold War and condemned China's 1989 crackdown on pro-democracy demonstrators in Beijing's Tiananmen Square.
Bill Clark CQ Roll Call
Allen says the band is honored to keep its promise to Bush to perform at his funeral.
Bush was hailed by presidential historian Jon Meacham as "America's last great soldier-statesman", who "made our lives and the lives of nations freer, better, nobler and warmer".
The service began at St. Martin's Episcopal Church, where Bush worshipped for more than 50 years, with more than 1,000 attendees singing "America the lovely". Jeb Bush and his wife joined along with about a dozen other members of the Bush family. He is the Texas land commissioner.
There, a motorcade will take Bush to his presidential library at the university, where he will be laid to rest at a private ceremony next to his wife, Barbara, who died in April, and his daughter Robin, who died at age 3 in 1953.
The Trumps visited shortly after Vice President Mike Pence delivered a moving speech about Bush and Pence's son Mike, who is a Marine aviator, that occurred only months before Bush passed away. Instead, they stood near the flag-draped coffin, with Trump saluting the 41st president while Melania held her hand on her heart.
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