"We almost made it big. We almost had a chance at power. We came that close, that close," Biden said during a breakfast with Irish Taoiseach Enda Kenny, according to a White House pool report. 

A number of American cardinals had been rumored to be in the running to become the next pope prior to the Vatican's selection of Pope Francis.  Among the most regularly mentioned was Cardinal Sean O'Malley of Boston.

Biden joked that O'Malley's communications director, Terry Donilon, who is the brother of White House Adviser Tom Donilon, "said something wrong."

Biden also shared his impression of Pope Francis during their first meeting.

"He seems like all he is advertised to be – a genuine, generous, humble man," Biden said.