Cole, who was one of eight players released by the club, joined the Galaxy in 2016 and went on to make 83 regular season appearances for the MLS side.
The defender had a base salary of $422,000 (£329,000) and a total compensation of $722,500 (£564,042), making him the second-best paid defender on a team that conceded 64 goals en route to missing the playoffs.
Cole was largely the bright spot in that defense, but, with the fullback now 37 years old, the club will go in a new direction.
The former England star isn't the only defender set to depart the club as Michael Ciani and Rolf Feltscher also saw options declined as the Galaxy move on from three of the four highest-paid members of the backline.
Goalkeeper Brian Sylvestre, forward Ariel Lassiter, defender Sheanon Williams, midfielder Baggio Husidic, and midfielder Servando Carrasco also saw options declined, while midfielder Chris Pontius, defender Dave Romney, and midfielder Emmanuel Boateng remain in negotiations with the club.
One name still on the roster is Zlatan Ibrahimovic, who is one of 16 players currently under contract with the club. Ibrahimovic has been linked with a move with AC Milan in recent weeks after scoring 22 goals in 27 games through his debut MLS season.
The Swiss star beat out Wayne Rooney to Newcomer of the Year honors while also winning Goal of the Season for his debut goal against Los Angeles FC.
The Galaxy had a chance to extend that season on the final day of regular season play, but collapsed against the Houston Dynamo by squandering a 2-0 lead en route to a season-ending loss that knocked the team out of the postseason race.