"Are you a good grandfather, do you play with them in the garden?" Stone follows up.
"Very seldom, unfortunately," Putin replies, with a seemingly sad smile.
Putin did not say how many grandchildren he has, nor which of his children had given birth.
Very little is known about Putin's daughters, who reportedly go by the names of Katerina Tikhonova and Maria Faassen.
According to media reports, Maria, the elder daughter, married Jorrit Joost Faassen, a Dutch businessman, and in 2013 Katerina married Kirill Shamalov, a Russian billionaire who owns a large stake in a gas and petrochemical company.
Putin has said that both his daughters, believed to be in their early 30s, now live in Moscow.
"I am proud of them," Putin told Stone. "They are not into big business or politics. They are into science and education."
Katerina is reportedly a senior manager at Moscow State University and competes in acrobatic rock'n'roll dance competitions. Almost nothing is known about Maria, who appears to be following a career in biomedicine.
In another interview with Stone, Putin says he almost refused to stand for president because of what he saw as danger to his children.
"I didn't know how long I could do it for because at any moment they could come along and say I was dismissed. And I could think about one thing only: where I would hide my children," he said.
Russian media is very careful about any information it publishes on Putin's family or alleged romantic ties.
In 2008, a Russian newspaper was shut down after it reported allegations of an affair between Putin and Alina Kabaeva, an Olympic gold medal-winning rhythmic gymnast 36 years his junior.
Asked about the allegation later that year, Putin denied it outright and criticised those who "with their snotty noses and erotic fantasies prowl into others' lives".
He announced in 2013 that he was separated from his daughters' mother, Lyudmilla Putina, after almost 30 years of marriage in a staged interview with a Russian journalist.
