Political opposites turned friends, former Presidents Bill Clinton and George W. Bush launched a new scholars program at four presidential centers with an opening act that might have been mistaken for a comedy routine.

The two former presidents -- one a Democrat, the other a Republican -- shared laughs and a buddy-like banter on stage Monday, talking about presidential leadership while trading stories about their famous families and life after the White House.

Bill Clinton said he and Bush laughed backstage about people coming up to them at restaurants and asking to take "selfie" photos. Quipped Bush: "At least they're still asking."

With Hillary Rodham Clinton seated in the fourth row, Bush noted that many people ask him about the possibility of another Bush-Clinton White House campaign. His father, President George H.W. Bush, lost to Clinton in 1992, and his brother, former Florida Gov. Jeb Bush, may seek the GOP nomination in a race that could pair him against Hillary Clinton.

"The first one didn't turn out too good," Bush quipped.

The 42nd and 43rd presidents joined together to announce the Presidential Leadership Scholars program, a partnership between the Clinton, Bush, George H.W. Bush and Lyndon B. Johnson presidential centers. At a time of partisanship and gridlock, both presidents said they hoped the scholars program would attract people in business, public service and the military interested in learning about presidential decision-making and applying it to their own careers.

Clinton revealed that he and Bush would speak twice a year during Bush's second term, 30-to-45 minute conversations about policy and politics. While they didn't always agree, Clinton said he never talked about their discussions and said the talks "meant a lot to me."

Clinton said the test of any democracy is finding ways of having a vigorous debate and still reaching resolution to the nation's problems. "If you read the Constitution, it ought to be subtitled: `Let's make a deal,"' Clinton said.

The elder Bush got in the act from afar, writing in a letter read aloud that every former president displays different qualities, "for example, not all of us skydive."

Bush, who celebrated his 90th birthday in June by making a tandem parachute jump, urged the two former presidents to "keep it brief."

The event had a light-hearted tone. Bush offered Clinton advice on becoming a grandparent -- Clinton's daughter Chelsea is expecting her first child this fall. "Get ready ... to be like the lowest person in the pecking order in your family," Bush said.

At one point, Clinton's cellphone rang on stage, prompting him to tell the audience that "only two people have this number and they are related to me." Clinton said he hoped "I'm not being told I'm about to become a premature grandfather."

Assessing each other's leadership qualities, Clinton said Bush did things he thought was right and "consistently benefited by being underestimated -- and so did I for totally different reasons." Bush said Clinton was empathetic and "an awesome communicator" who could "really lay out a case and get people all across the political spectrum." Ending his comments, he asked Clinton: "Is that enough?"

George W. Bush campaigned for president in 2000 on restoring "honor and dignity" to the White House following Clinton's impeachment over a sex scandal. But the two former presidents have developed a bond, strengthened by their mutual admiration for the elder Bush, whom Clinton visited in Maine last week.

Clinton and the younger Bush worked together on relief efforts after Haiti's devastating earthquake in 2010 and have been active in the fight against HIV/AIDS in Africa.

Bush noted that his upcoming book, called "41: A Portrait of My Father," would be a "love story. It's a story about seeing someone you admire and learning from them."

As moderator Josh Bolten humorously plugged the book's Nov. 11 release date, Clinton mused about writing his own competing Bush 41 book. "I think I can put one together that would be ready to go," Clinton joked.