Mr Bush, who served as the 41st U.S. president between 1989 and 1993, died late on Friday at the age of 94.
Sully will reportedly be traveling with the former President's casket on his flight to Washington, D.C. on Monday.
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.
"I think he was one of the kindest, most generous men", said Marge Frazier, who visited the downtown statue on Sunday while showing friends from California around. 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.
Vice President Pence, remembering Bush's service as a naval aviator in World War II and his life in public service, said "he never failed to answer the call to serve his country". The Facebook page for the George H.W. Bush Presidential Library and Museum also shared the image.
After making the short trip back to the White House, Trump posted a video to Twitter with the hashtag #Remembering41.
Bush's body will be transported to the U.S. Capitol, where he will lie in state in the building's rotunda until Wednesday morning. In August, the late GOP Sen. John McCain was the last public figure to lie in state there.
Bush will then lie in repose at St. Martin's Episcopal Church in Houston, where first lady Barbara Bush's funeral was held in April.
Trump has ordered the federal government to close on Wednesday and both the New York Stock Exchange and Nasdaq will be closed in observance.
Mourners began lining up at the Capitol to pay respects to Bush, who died at his Houston home on Friday night at the age of 94, seven months after the death of his wife Barbara.
Following the funeral in Washington, Bush's casket will be returned to Houston, where a private service is scheduled for Thursday.