George W. Bush and Bill Clinton may not agree on much, but the two have developed a post-presidential friendship.

Onstage together at the Newseum today, they bonded over something they'll soon have in common: being grandfathers.

About 20 minutes into the event promoting the Presidential Leadership Scholars program, President Clinton's phone rang.

"Only two people have this number and they're both related to me," Clinton quipped. "I hope I'm not being told I'm about to become a premature grandfather!"

"That'd make national news," Bush responded, as the audience laughed.

Clinton's only daughter, Chelsea Clinton, recently announced she and husband Marc Mezvinsky are expecting their first baby this fall. Bush's grandchild, Mila Bush-Hager, was born in April 2013.

Chelsea Clinton: I'm Pregnant!

Will Clinton Baby Affect 2016, and Is It Sexist to Ask?

What Will George W. Bush's Granddaughter Call Him?

Bush offered some advice for Clinton as a grandfather, once he becomes one.

"Be prepared to fall completely in love again," Bush said. "It's going to be an awesome period."

He also joked, "Get ready to be, like, the lowest person in the pecking order."

The two engaged in a nonstop lovefest onstage, with Bush jokingly describing his predecessor as a "beautiful man," when asked to comment on him.

At the end, as the two made their way to the exit, working the crowd and shaking hands, former First Lady and Secretary of State Hillary Clinton (who was in attendance) greeted her husband and George W. Bush.

"I actually learned a lot by watching him over the years," Bill Clinton said of Bush onstage.

"You always want to be underestimated by your adversaries. He consistently benefited from being underestimated, and so did I, for comets my different reasons," Clinton joked.

Bush called Clinton his "pal" and praised him as an "awesome communicator."

Watch more news videos | Latest world news

Clinton also revealed that the two spoke over the phone periodically during Bush's second term.

"He used to call me twice a year on his second term, just to talk," Clinton said. "We talked about everything in the wide world."

Bush would ask Clinton's opinion, and often they'd disagree, Clinton said.

Bush and Clinton were onstage together at the Newseum to talk about presidential leadership, launching a new joint scholarship program on that topic, which will be run through the Clinton Foundation, Bush's presidential library, and the presidential libraries of George H.W. Bush and Lyndon B. Johnson. The program will bring in experts and past administration officials to and help participants study presidential leadership and decision-making.

The two have seemed to enjoy a friendly relationship. They appeared onstage together for the first time in 2009, defending each other during a forum in Toronto, the AP reported at the time.

More recently, Bush nominated Clinton to participate in the ALS ice-bucket challenge, in a video that featured Laura Bush pouring ice water on the ex-president's head. Clinton has yet to accept the challenge.

ABC's Erin Dooley contributed to this report.