Auction house Sotheby's is offering a working draft of the finished song in Dylan's own hand for an estimated 1 million to 2 million US dollars (£600,000 to £1.2 million).
The draft is written in pencil on four sheets of hotel stationery and contains revisions, additions, notes and doodles.
Scrawls seem to reflect the musician's experimentation with rhymes.
Dylan was only 24 when he recorded the song in 1965.
The song is about a debutante who becomes a loner when she is cast from upper-class social circles.
The doodles include a hat, a bird and an animal with antlers, while the stationery comes from the Roger Smith hotel in Washington, DC.
"How does it feel To be on your own" it says in Dylan's handwriting. "No direction home Like a complete unknown Like a rolling stone."
Scrawls seem to reflect the artist's experimentation with rhymes.
The name "Al Capone" is scribbled in the margin, with a line leading to the lyrics "Like a complete unknown."
Another note says: "...dry vermouth, you'll tell the truth..."
In 2010, John Lennon's handwritten lyrics for A Day In The Life, the final track on the Beatles' classic 1967 album Sgt. Pepper's Lonely Hearts Club Band, sold for 1.2 million US dollars (£709,000), the record for such a sale.
Sotheby's described the seller as a long-time fan from California who "bought directly from Dylan". He has not been identified.
The auction house said it is "the only known surviving draft of the final lyrics for this transformative rock anthem".
The sale is taking place on June 24.