The 46-year-old, in his second spell in charge at the Priestfield Stadium, had his contract terminated on Tuesday afternoon following a meeting with chairman Paul Scally.
He has, however, been offered a place on the board and a football development role at the club where he also spent 10 years as a player.
A club statement read: "The chairman would like to thank Andy for all his hard work and effort over the past two years and both he and Andy, while disappointed the club has not achieved its aims this season or last, want nothing but success for the club going forward.
"Given the huge and invaluable contribution Andy has made to the club, the chairman has asked him to consider joining the board as a full company director in a role that would assume responsibility and oversee all club football development, scouting, recruitment and all other football related matters.
"Andy has agreed to consider this offer over the next few days.
"In light of the above, (coach) Nicky Southall will be leaving the club forthwith. We wish him every success in the future."
Gillingham finished in eighth place this season, two points outside the play-off spots.