The pontiff had invited the couple to a seminar he was giving at the Vatican on Sunday.
During the event Francis ruled out following in the footsteps of his predecessor Emeritus Pope Benedict XVI and resigning the papacy.
Responding to a question he said: "I never thought of quitting being pope, or of leaving because of the many responsibilities."
Francis had gathered a number of high profile people for his 'Un Muro o Un Ponte' seminar including Salma Hayek and Richard Gere.
Clooney was awarded a medal by the pope for his role in using schools to promote social integration.
Before the event Francis again raised the plight of refugees coming to Europe for sanctuary.
Refugees "aren't a danger but they are in danger", he said.