Clinton added he wanted to see frequent debates that allowed voters to see how the candidates would shape policy to tackle complicated issues.
"So much in the media today is this culture of anger and resentment. We have to rise above anger to answers, to rise above resentment to a real response," he said.
Clinton also told the audience "the world belongs to tomorrow, not yesterday.
"Someday, for all of us, it will be our last day. What will matter will be all the steps we took along the way and what they amounted to," he said.
Bush responded: "I was stuck on that 'some day may be your last day' line. That was pretty damned profound."
Bush talked about the need to avoid the trappings of power and create an environment around leadership where "sycophants are not allowed in," adding Clinton told him to use big words.
Clinton replied that he did not know any big words, except perhaps "itinerant portrait artist," apparently referring to Bush's hobby of painting portraits since retirement.
The Presidential Leadership Scholars program is a partnership among the Bush, Clinton, Lyndon Johnson and George H.W. Bush presidential centers.