For starters, they are both World War II heroes and they both ran for president in the 1980s, with Bush finally prevailing in 1988.
Dole, 95, appeared at the Capitol rotunda on Tuesday to pay his respects to the late president.
During Dole's salute to Bush, the attending crowd remained silent.
Bush achieved the office that Dole sought in 1996 as the Republican presidential nominee.
Bush will be commemorated at a service at the National Cathedral Wednesday morning before being laid to rest Friday in Texas.
President Donald Trump has declared December 5 as a National Day of Mourning.
As a specially-trained service dog, Sully was able to assist Bush, who was suffering from Parkinson's in his later years, by opening doors and being able to get help, should the former president have needed it.
Michelle Obama has once called George Bush her "partner in crime" at every major event where all former resident of the White House gather to highlight the importance of their friendship for her.More news: Trump keeps up pressure on Dems over border wall
In an invocation opening proceedings, the US House chaplain, the Reverend Patrick J Conroy, praised Mr Bush's commitment to public service, from Navy pilot to congressman, United Nations ambassador, envoy to China and then Central Intelligence Agency director before being elected vice-president and then president - the first incumbent to do so in over 100 years. Among them was the former US President's prominent friend - Sully, his service dog.
His wife, Elizabeth, is also very close with the Bush family, and served as a Republican senator from North Carolina for several years.
"Here lies a great man", said Rep. Paul Ryan, the House speaker, and "a gentle soul".
After Thursday's private services, the former president will be taken by a train, pulled by an engine commissioned in his honor, 120 kilometers north to the city of College Station for burial at his presidential library on the grounds of Texas A&M University.
The federal government and some financial exchanges will be closed on Wednesday for a day of mourning.
Flags on public buildings are flying at half-staff for 30 days.
George W. Bush greeted Melania Trump with a kiss outside Blair House.