The dog is named after the airline pilot Chesley "Sully" Sullenberger, who landed a passenger jet on the Hudson River in 2009, saving all 155 passengers and crew on board.
The casket will be accompanied to Joint Base Andrews, outside Washington, by the 43rd president, Bush's son George W Bush; his wife, Laura; his brother Neil Bush and his family.
"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 he brought to 41", the younger Bush wrote.
The body of former US President George HW Bush has arrived at the US Capitol in Washington, where mourners will pay tribute to the man who led the nation as the Cold War ended.
"As one person said, he can do just about anything except make you a martini, but not to worry, he can go get you someone to make you a martini", Bush family spokesman Jim McGrath said in June. His fans have multiplied since his work gained more attention following Bush's death, and an outpouring of appreciation for the dog has been part of the mourning process.
He died in Texas, his home state, late Friday at the age of 94 after several years of failing health.
Political analyst Larry Sabato at the University of Virginia said Bush's stature among the country's 45 presidents has grown in the quarter century since he left office. The casket was flown in from Ellington Field Joint Reserve Base, a Texas Air National Guard base, aboard an aircraft that often serves as Air Force One.
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.
Sully, who had been in the former President's service since early this year, was assigned to him in his last days in a wheelchair.
"Our thoughts and prayers are with the entire Bush family during this hard time", said John Miller, president and CEO of America's VetDogs.
Trump has ordered the federal government to close on Wednesday and both the New York Stock Exchange and Nasdaq will be closed in observance.