The actor, in character as his ruthless House of Cards politician Frank Underwood, mimics his pal Clinton in a call to Hillary, asking her if he can have an elephant for his birthday.
When she insists such a creature might be dangerous for the family's dogs, Spacey - as Underwood impersonating Bill - says, "What about a baby elephant?"
He goes on to suggest names for their daughter Chelsea's baby, offering up Frank if it's a boy and Claire if it's a girl - Robin Wright plays Frank Underwood's scheming wife Claire in House of Cards.
Hillary jokes she was thinking of the name Kevin, urging Spacey to hang up and simply sign her husband's online birthday card.
Bill Clinton turns 68 on 19 August (14).