Sir Alex was offered the Gunners' job in 1986 and was planning on taking Walter Smith with him to Highbury as his assistant.
With Scotland preparing for the up-coming World Cup in Mexico, however, the Gunners refused to allow him to take the post after the tournament had finished resulting in a breakdown in talks.
Sir Alex revealed in the matchday programme for Rangers' clash with Dundee United: “I was offered the Arsenal job and I wanted to take Walter with me.
“I thought I would sound him out when we were on Scotland duty together. We were going to play in Israel and I didn't know that Walter was going to Rangers at that time.
“I told him that I had been offered the chance to go to Arsenal and I asked him what he thought. He was positive about it, saying they were a big club and that kind of thing.
“So I said to him 'Do you fancy coming with me?' Then he dropped the bombshell that he was going to Rangers.
“The thing about the Arsenal offer was that they wanted an answer right away but I couldn't give them one because I was going to the World Cup in Mexico with Scotland.
“So the Arsenal thing fell by the wayside. Walter went to Rangers and then later that year I went to Manchester United.”