Bush's son George W. Bush, the nation's 43rd president, posted the photograph on Instagram, with a message announcing Sully's transfer to Walter Reed National Military Medical Center in suburban Maryland.
Maine Governor Paul R. LePage called President Bush "Maine's president" in an order closing executive departments in the state, where the Bush family has a summer home.
People with knowledge of the funeral plans for former President George H.W. Bush reportedly say the proceeding will focus on his life, rather than becoming a politically-charged event.
The honorary pallbearers in Washington - for both Bush's entry into the Rotunda and the National Cathedral - will be Navy captains and admirals who have commanded the USS George H.W. Bush and members of the carrier strike group.
Search Underway After Two US Marine Aircraft Crash Off Japan
The crash comes just a month after a US F/A-18 Hornet from the aircraft carrier USS Ronald Reagan crashed into the sea southwest of Okinawa.
In a statement on Saturday, Trump again referenced the Bush's "thousand points of light" speech and said the former president "inspired generations of his fellow Americans to public service".
Former Canadian Prime Minister Brian Mulroney praised Bush as a strong world leader who helped oversee the end of the Cold War and the collapse of the Soviet Union and set the stage for the North American Free Trade Agreement with Canada and Mexico, achieved under his successor, Clinton.
But late Monday, after Bush relatives and dignitaries had gone, the president was driven from the White House up to Capitol Hill.
President from 1989 to 1993, Bush died Friday at age 94.
That Sully will continue to serve veterans was important to the Bush family. Trump himself has criticized the elder Bush on the campaign trail. Flags on public buildings are flying at half-staff for 30 days.
McGrath also tweeted a photo of Sully, Bush's service dog, laying in front of the late President's casket on Sunday night with the caption, "Mission Complete". President Bush will be laid to rest Thursday at his presidential library on the Texas A&M University campus, where his wife Barbara and their daughter Robin are also buried. He had joined the Navy when he turned 18. He would go on to serve as ambassador to the United Nations and China, head of the Central Intelligence Agency and chairman of the Republican National Committee before being elected to two terms as Ronald Reagan's vice president.