Chapman "wanted to be friends" with the cop who was first on the scene at the 1980 shooting and said they were "part of something big".
Lines from the first of three typed letters to Spiro in 1983 reads, "Besides wanting to be your friend, is there any way that you can help me locate my copy of 'The Catcher in the Rye' that was taken from me on the night of my arrest?"
Chapman was infamously collared with a copy of the JD Salinger novel inscribed with, "This is my statement".
The letters are being sold through Gary Zimet's Moments in Time and valued at USD 75,000.
"From the time of my arrest I have felt close to you... It is something that would happen to Holden Caulfield," Chapman wrote in another letter.
"We were all part of something big, Steve. Something phenomenal. It is all going to come out," he added.