The conversations-- which took place between 1998 and 2000--were secretly taped by Doug Wead, a former aide to Bush's father, who has now revealed some of the recordings.
In the tapes, Mr Bush mocks Al Gore, his opponent in the 2000 elections, for admitting to smoking the drug.
On the tape, the future president discusses his refusal to answer questions about whether he took marijuana, saying any such admission might affection his standing if he won the presidency.
"I wouldn't answer the marijuana question," he says.
"You know why? 'Cause I don't want some little kid doing what I tried."
He feared that any such admission might affection his standing if he won the presidency.
"Do you want your little kid to say 'Hey daddy, President Bush tried marijuana, I think I will'? he asked.
As well as mocking then Vice-President Al Gore for admitting to smoking the drug, Mr Bush is heard dismissing him as a pathological liar.
According to The New York Times, 'Wead said he recorded the conversations because he viewed Mr. Bush as a historic figure, but he said he knew that the president might regard his actions as a betrayal. As the author of a new book about presidential childhoods, Mr. Wead could benefit from any publicity, but he said that was not a motive in disclosing the tapes.'
The White House has not disputed the authenticity of what a spokesman said were "casual conversations".
"These were casual conversations that then Governor Bush was having with someone he thought was a friend, and that's what they are," said White House spokesman Scott McClellan.