Albright, Clinton recalled, assured him in that conversation about two weeks before she died last month, that she was getting the best care she could but didn't want to "waste time" talking about that.
"The only thing that really matters is what kind of world we're going to leave to our grandchildren," Clinton recalled Albright told him.
He added, "She made a decision with her last breath she would go out with her boots on".
Led by US President Joe Biden and former presidents Barack Obama and Clinton, the man who picked Albright to be his top diplomat and the highest-ranking woman ever in the US government at that time, mourners gathered to celebrate her life and accomplishments of the child refugee from war-torn Europe who rose to become the first female US secretary of state.
Albright died of cancer last month at age 84, prompting an outpouring of condolences from around the world that also hailed her support for democracy and human rights.
Besides the current and former presidents, the service was attended by at least three of her successors as secretary of state along with other current and former cabinet members, foreign diplomats, lawmakers and an array of others who knew her.
Biden, who delivered a tribute to Albright, said her name was synonymous with the idea that the United States is "a force for good in the world".
"In the 20th and 21st century, freedom had no greater champion than Madeleine Korbel Albright," Biden said.
"Today we honour a truly proud American who made all of us prouder to be Americans."
Hillary Clinton urged her husband to tap Albright to serve as secretary of state, a role that Clinton would serve in herself during Obama's administration.
The two developed a strong friendship over the years.
Clinton in her own tribute recalled some lighter memories of Albright, including the time she taught the foreign minister of Botswana the Macarana and danced the night away with a young, handsome man at her daughter Chelsea's wedding.
She also remembered Albright as a fearless diplomat that broke barriers and then counseled, cajoled and inspired women to follow in her footsteps.
"The angels better be wearing their best pins and putting on their dancing shoes," Clinton said.
"Because if as Madeleine believed there's a special place in hell for women who don't support other women, they haven't seen anyone like her yet."
The crowd of about 1400 that gathered at Washington National Cathedral to honour Albright included the current secretary of state, Antony Blinken, and former secretaries Condoleezza Rice and John Kerry were slated to attend.
Other top current officials expected to be present included Defense Secretary Lloyd Austin, CIA Director Bill Burns, Joint Chiefs of Staff Chairman General Mark Mark Milley and White House national security adviser Jake Sullivan.
The members of the VIP audience were masked, as Albright's family had requested.
Foreign dignitaries invited to the funeral included the presidents of Georgia and Kosovo and senior officials from Colombia, Bosnia and the Czech Republic.
