Naismith had trials at Arsenal as a teenager.
He told the Scottish Sun: "Up until going to Arsenal I'd been playing for the team that was most local to me, staying at home and all the rest of it. When Jim Jefferies came in and mentioned about going I was in shock.
"I didn't expect it. But I went down and had a few great days. Meeting Arsene Wenger and his assistant Pat Rice was something in its own right.
"Then when I walked into the dressing room Ashley Cole and Theo Walcott sat there.
"I think they were impressed with me as a person as well as what I did on the training pitch.
"I think I coped with it all right but I'd definitely say I wasn't ready to move at that stage.
"Nothing came of it which was maybe a good thing because I wouldn't have liked to have had to have made a decision like that at that stage of my career. I had another good couple of years at Kilmarnock which helped me mature a bit more.
"It opened my eyes to the bigger picture. The facilities were world class, the manager spoke a lot of sense and it was amazing. But I wouldn't have said it was the right time to move then. I hadn't really done a lot in the game to warrant one."