Moyes, who is now coach of Real Sociedad after an unsuccessful season in charge at Old Trafford, says that he was approached by then-Gunners boss Terry Neill over a switch to Highbury.
But the Scot instead decided to join Cambridge United after the north Londoners only decided to offer a loan deal.
Asked if it was true that he almost joined Arsenal, Moyes told FourFourTwo: "It is. When I tell people they start laughing.
"Cambridge were in what's now the Championship. Arsenal manager Terry Neill called me at my house in Glasgow and told me they were going to sign me but, a week later, they called back and said: 'We've decided to take you on loan.'
"I wasn't having that, leaving Celtic to go on loan, so I turned it down. Arsenal went on to buy the late Tommy Caton instead. I joined Cambridge and played every week, which I wasn't doing at Celtic."
