I take it this is when playing the open E? How hard are you hitting the string? How close is the string to the top of the first fret, perhaps the nut slot is cut a bit too deep?
The nut slot can be built up so a new nut may not be necessary. Do some searching on the forum, I know someone has a good explanation on what he used to do so. As a temporary fix you can put a piece of paper or something in the slot to raise the string up a bit. All four slots on my old '67 4005WB had slots that were worn down that I had folded pieces of paper in them to prevent buzzing; it stayed that way for many years and worked just fine, regardless of what others may say. I "painted" the paper black with a Sharpie.