Speaking on the second day of the Group of Eight summit in the French spa resort of Evian-Les-Bains, Bush said: "I know there are a lot of people in both our countries wondering whether or not we can actually sit down and have a comfortable conversation. And the answer is absolutely."
In the tranquil setting of the Hotel Royal on the banks of Lake Geneva, Bush and Chirac made a conspicuous effort to patch up relations damaged by the conflict in the Persian Gulf.
Putting his hand on his host's shoulders, Bush said, "We can have disagreements but that doesn't mean we have to be disagreeable to each other."
He added: "Listen, we must be frank, we went through a difficult period. I understand his position. He made it very clear to me in the very beginning. There was no question where Jacques Chirac stood, and I made it clear where I stood. That's why I can say we have good relations, because we're able to be very honest with each other."
In an attempt to break the ice, Bush offered the 70-year-old statesman a set of books on American Indian civilization. Chirac's spokeswoman Catherine Colonna said the French president was "particularly touched by the gesture" and that Bush's gift had "hit it right on the spot."
An unusually subdued Chirac described the meeting as "very positive," and wished Bush well on his Middle East peace mission, which he said France fully supported.
Before leaving Monday to meet Arab leaders in the Egyptian resort of Sharm el-Sheik, Bush acknowledged talks between Israelis and Palestinians would be "difficult" but pledged to "put in as much time as necessary to achieve the vision of two states living side by side in peace."
To underline the new spirit of cooperation between Paris and Washington, Colonna announced France would send special forces to work alongside U.S. troops in Afghanistan. She also said Chirac and Bush shared the same concerns about Iran's nuclear weapons program.
Despite the somewhat forced attempt to patch up relations, new fault lines emerged during discussions Monday after France reportedly voiced concerns about U.S.-sponsored plans to prevent the proliferation of weapons of mass destruction by sea and air.
The leaders of Britain, Canada, France, Germany, Italy, Japan, Russia and the United States are also expected to struggle to find common ground on ways to boost economic growth and help developing nations during a summit shaken by violent clashes between police and anti-globalization protesters.
Celebrity Breakups and divorces of 2014 [PHOTOS]