"A great joy to welcome home the newest member of our family, 'Sully, ' a lovely, and beautifully trained, lab from America's VetDogs".
He also credited Bush, who died Friday night at age 94, with working with other countries and the United Nations in the first Gulf War.
From there, the president's casket will be taken to the U.S. Capitol in Washington.
The casket carrying the body of George HW Bush is set to arrive in Washington on Monday afternoon, aboard the United States military plane commonly known as Air Force One.
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.
The picture was tweeted by Mr Bush's spokesman, Jim McGrath, showing Sully next to Mr Bush's casket on Sunday along with the caption: "Mission complete". And beyond personality, the conflicting presidencies underscore just how little remains today of the Grand Old Party that Bush once led.
He will be interred at the family plot at the George H.W. Bush Presidential Library and Museum in College Station. Wednesday has been designated by President Trump as a national day of mourning, and most federal agencies will be closed with federal workers given the day off.More news: Urban Meyer stepping down at Ohio State, Ryan Day to take over
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Memorial services were underway Monday to honor the late President George H.W. Bush, whose casket is set to depart Houston for a public viewing at the U.S. Capitol until Wednesday and a ceremony at the National Cathedral in D.C.
The dog has also accompanied the family during the ceremonies marking Bush's passing. The crew has been tasked by Trump with carrying out "Special Air Mission 41".
"America's VetDogs is deeply saddened by the loss of President @GeorgeHWBush". Earlier Monday, Trump had posted that he was "Looking forward to being with the Bush Family to pay my respects to President George H.W. Bush".
Almost three decades later, Bush's leadership approach stands in sharp contrast with that of President Donald Trump, a fellow Republican whose brand is defined by material success, unrestrained aggression toward his rivals and disdain for traditional coalitions at home and overseas.
Asked about his call on Saturday with President Trump, he said it was a brief, but pleasant conversation.
The last funeral for a former president was the one for Gerald Ford, who died on December 26, 2006 and was buried January 2; Ronald Reagan had died two years earlier in June 2004.