While I recall several Economists saying that this connection exists, it has never happened and Greenspan, to my knowledge has never stated it in this way.
Someone correct me if I am wrong.
About the only link you could draw is that long term interest rates declined as the deficit fell in the mid 1990's.
But our deficits are still fairly low compared with the size of the economy, so the impact is minimal. However, if we are running trillion dollar deficits in the next few years then you would likely see an impact in interest rates. What is more disconcerting is the fact that over half of our annual deficit is being bought by the governments of China and Japan. China could cripple our economy if it were to decide to float a large chunk of our debt and instead buy debt in other countries. To continue to sell more debt we'd have to jack up the interest rates that we offer on Treasury bills.
I would think that interest rates reflect the risk of not getting all my money back. I guess another factor if I'm loaning money is if I can get a better return somewhere else.