Sully in front of George H.W. Bush's casket on Sunday.
This week will see four days of events including a state funeral at the National Cathedral in the U.S. capital on Wednesday.
Bush spokesman Jim McGrath posted a picture on Sunday of the yellow Labrador retriever named Sully with the caption, "Mission complete".
Trump ordered flags on the White House and public buildings flown at half-staff - an honour he did not initially accord Senator John McCain - and proclaimed Wednesday a day of national mourning.
"As much as our family is going to miss this dog, we're comforted to know he'll bring the same joy to his new home, Walter Reed, that be brought to 41", Bush, the 43rd president, said.
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Former President George H.W. Bush wears socks with his image as he sits in a cart on the sidelines before an NFL football game in Houston, Dec. 1, 2013.
Former President George H.W. Bush will be eulogized by his son, former President George W. Bush, along with a mix of family and friends on Wednesday at the Washington National Cathedral, the most high-profile event in a week of proceedings that will remember the remarkable life of the president who died on Friday at 94.
Sully will now make the final journey with Bush's casket to Washington, DC, on Monday afternoon a source told CNN, where the 41st president will lie in state at the US Capitol from 5 p.m. on Monday until 10 a.m. on Wednesday.
Shortly before leaving the service, he married his 19-year-old sweetheart, Barbara Pierce, and forged a 73-year union that was the longest presidential marriage in USA history until her death.
He will be buried at the presidential library in Texas, alongside his wife, Barbara Bush, who died seven months ago.
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Abigail Ogle, an Oklahoma City local news anchor, tweeted: "Sully, the service dog of President George H.W. Bush, keeping watch over his best friend".
Former prime minister Brian Mulroney, whose nine years in power overlapped with Bush's four, said in a written statement that he and his wife share in the Bush family's grief. Those were the ones Bush chose, and the proceeds from the socks went to the Barbara Bush Foundation for Family Literacy. He would go on to serve as ambassador to the United Nations and China, head of the Central Intelligence Agency and chair of the Republican National Committee before being elected to two terms as Ronald Reagan's vice-president.
His love of socks also led to some forged connections, including one with an entrepreneur with Down syndrome who started his own sock company.
"I invite the people of the world who share our grief to join us in this solemn observance", he said in a December 1 proclamation.
Bush replied: "Good, that's where I want to go".
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Baker said Bush's last words, in a phone conversation with son George W. Bush, were these: "I love you too".