"I hope by the first of October, I'll be a grandfather," he told CNN. "I can't wait. We're on watch now."
Bill Clinton and wife Hillary are awaiting the arrival of their first grandchild
Chelsea, 34, and her investment banker husband Marc Mezvinsky announced in April they were expecting their first baby together.
Bill also revealed that the couple have chosen not to find out the sex of the child.
"I don't know. My daughter and son-in-law decided not to know. They want to be surprised," he said, when asked if the new arrival is a boy or girl.
His wife Hillary Clinton similarly admitted the couple were on "grandbaby watch" while attending a woman's event last week.
Chelsea Clinton and her husband Marc Mezvintsky
"Now I've been thinking a lot about family because you know I'm on 'grandbaby watch' and I think a lot about this new member of our family and what he or she can look forward to," the former secretary of state said.
"Whether you're the grandchild of a president or the grandchild of a janitor, whether you're born in a city or in a small rural village – no matter who you are, you have the right to inherit the American dream."
Chelsea and Marc started dating in 2005, and tied the knot in a traditional and lavish ceremony in the state of New York in August 2010.
After announcing her pregnancy, Chelsea, 34, said she hoped to "be as good a mum to my child, and hopefully children, as my mum was to me".