On Sunday night Thompson instructed lawyers to act on his behalf after being told of his dismissal by manager Neil Lennon over the telephone.
Celtic claimed Thompson, 38, failed to attend a pre-arranged meeting with the Parkhead boss in Newcastle.
A statement, released on Tuesday evening through the player's solicitors, Bridge Litigation, read: "Alan Thompson has been dismissed by Celtic Football Club after enjoying years of first-class football both as a player and first-team coach.
"This included playing at the very highest level, both domestically and in Europe, then coaching the first team to become league champions.
"He is profoundly disappointed, given his strong feelings for the club, but remains deeply proud of his long and successful association with Celtic.
"Legal action has been instructed and is at a preliminary stage, however, Alan wishes this all be to be resolved amicably with Celtic, given his acknowledged feelings for the club."