The dog was named after the pilot who landed a damaged passenger plane on the Hudson River in New York City in the 2009 "Miracle on the Hudson" incident.
The organization said Sully will be joining Walter Reed National Military Medical Center's facility dog program following his service with the late president.
George H.W. Bush is set to embark on his final tour of Washington, the USA capital that is remembering the 41st president's lifetime of public service that began in the navy during the Second World War, ended with one term as the country's leader, and was characterized by what admirers say was his innate decency, generosity and kindness.
"He kept us flying high and challenged us to fly higher still, and he did it with modesty and kindness that would've been surprising in someone one-tenth as tough and accomplished as he was", McConnell said.
Sully worked for Bush over the summer after former First Lady Barbara Bush died.
Bush received the two-year-old Labrador in June from America's VetDogs charity. H.W. Bush will be honored at several public and private events in Houston and Washington before his burial on Thursday in Texas.
In Texas, students, staff and visitors have been flocking to Bush's presidential library on the campus of Texas A&M University.
He said Bush was the first president he ever voted for and was "the first president to teach me and many of us that in a democracy sometimes you fall short", and how you handle that, "is just as important as how you win". Bush's family has not said who will speak at the service. His remains are scheduled to arrive at Andrews Air Force Base in Maryland in the afternoon.
The body of former President George H.W. Bush will lie in state in the U.S. Capitol on Monday as his family and Americans mourn the loss of the 41st U.S. president.
The 41st president of the United States, who lived longer than any of his predecessors and possessed one of the most impressive resumes in American political history, died peacefully at his home in Houston, according to a source close to the family.
It features a photograph of Sully laying down in front of Bush and Bill Clinton, the man who defeated him in the 1992 presidential race.